System and method for measuring characteristics of a continuous medium and/or localized targets using multiple sensors

ABSTRACT

A method and system is provided for obtaining data indicative of at least one characteristic of a continuous medium or at least one localized target located within a predetermined volume of space. The system includes a sensor configuration and a processing circuit. The sensor configuration includes a plurality of sensors for acquiring a plurality of signals from the continuous medium or the target(s). The plurality of sensors have centers spatially separated from each other in at least one spatial dimension. The processing circuit is configured for obtaining data indicative of the characteristic or characteristics of the medium or the target by calculating a plurality of powered weighted increments using the plurality of signals acquired by the sensor configuration and by using a plurality of models for relating the plurality of powered weighted increments to the characteristic or characteristics of the medium or the target. The selected characteristics of the medium or the target can be estimated with better accuracy and temporal and/or spatial resolution than is possible with prior art correlation function, spectra, and structure function-based methods and systems. In contrast to prior art methods and systems, the invention is not sensitive to signal contaminants with large temporal scale such as ground and sea clutter, and is not sensitive to low frequency external interference. The invention is capable of estimating various characteristics of the medium or target, for example, size, shape, visibility, speed, direction of the motion, and rates of changes of the above characteristics.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to the field of detection and monitoring andspecifically to a system and method for measuring the characteristics ofcontinuous medium and/or localized targets in a predetermined volumeusing multiple sensors.

2. Description of the Related Art

An important problem in the field of detection and monitoring isaccurately and reliably measuring as many characteristics of a monitoredobject as possible. Adequate accuracy and reliability are especiallyimportant in the case of remote detection and monitoring under adversemeasurement conditions. Characteristics of a monitored object producedby detection and monitoring equipment seldom represent the final productof a measurement system. Typically these characteristics are furtherinterpreted for making decisions and/or recommendations and thedecisions and/or recommendations are presented to the system's users. Byreliably and accurately measuring more characteristics of the monitoredobject, one ensures more robust and definite decision making andsignificantly decreases the probability of making an incorrect decision.Best possible data interpretation is especially important for automatedreal-time systems.

The monitored object can be a predetermined volume in a continuousmedium such as the atmosphere, lakes, rivers, the ocean, surface andsubsurface terrain, the human body, a chemical reactor, or any othermedia. Measured characteristics of such medium in a specified volume areused, for example, in the fields of meteorology, weather forecasting,geology, agriculture, medicine, and astronomy. Additionally, measuredcharacteristics of such medium in a specified volume are also used, forexample, in monitoring the airspace around airports, in monitoringconditions in chemical and processing plants, and in monitoring othersomewhat similar processes and physical configurations. Monitoredobjects can also be specified targets located in a predetermined volume,such as, missiles, airplanes, obstacles, defects in a product,intruders, or other specified targets; in these cases the measuredcharacteristics of the targets are used for purposes of national defenseand homeland security, collision avoidance, non-destructive producttesting, business and personal protection, and the like. It should benoted that in the instant disclosure the general term “monitored object”can be construed to refer to a predetermined volume in a continuousmedium or a specified target in a predetermined volume. When a specifiedtarget is located in a predetermined volume it may also be referred toas a localized target.

Existing monitoring equipment can be divided into two classes namely,single sensor equipment and multiple sensor equipment. Single sensorequipment, such as, for example, standard single-receiver Doppler radarsand individual pressure, temperature or other in-situ probes have beenand are still widely used for numerous applications. However, suchequipment provides a relatively small amount of initial informationabout the monitored object. This factor significantly limits the numberof characteristics of the object that can be determined and also limitsthe accuracy and reliability of measurements, especially at adverseconditions. Manufacturing and implementing multiple sensor monitoringequipment capable of performing real-time operations is now possiblebecause of the outstanding progress in electronics and computertechnologies that has been made during the last several decades. Whencompared with single sensor monitoring equipment, multiple sensormonitoring equipment provide a dramatically larger amount of initialinformation about the monitored object and therefore, enable morereliable measurement of a greater number of characteristics of themonitored object with a much higher accuracy under any conditions. Knownexamples of multiple sensor configurations include arrays of receivingantennas used in spaced antenna radars, arrays of microphones used forunderground exploration or for detecting airplane wake vortices, andarrays of in-situ temperature probes used for measuring characteristicsof convective or reacting flows.

Monitoring equipment that obtains signals from multiple sensors producesa large amount of initial information about a monitored object incomparison to the information produced with equipment that obtainssignals from only a single sensor. The objective of data analysis is toaccurately and reliably extract as many useful characteristics of theobject as possible. All data processing methods using data obtained frommultiple sensors are basically similar in that they utilize the sameinitial information: a time series of signals from a plurality ofsensors. The methods differ by the mathematical functions used foranalyzing the signals, the mathematical models for relating thesefunctions to the characteristics of the monitored object, and theassumptions that are adopted for constructing the models.

Traditional correlation function and spectra-based data processingmethods for multiple sensor monitoring equipment have been widely usedfor decades in numerous areas of applications. At the same time, thedrawbacks of these methods have been well recognized and thoroughlydocumented. They are as follows: (1) a poor temporal and/or spatialresolution; (2) the inability to operate in adverse measurementconditions such as external interference and strong clutter; (3) a lowreliability of measurements due to the adoption of inappropriate, oftentoo restrictive assumptions, and (4) a limitation in the number ofcharacteristics of the monitored object that can be retrieved. Forexample, the variance of the vertical turbulent velocity is the onlycharacteristic of atmospheric turbulence that can be retrieved with aspaced antenna profiler using traditional data processing methods.

Drawbacks of traditional data processing methods described hereinabovehave been partly addressed by the structure function-based method. Inthe particular case of a spaced antenna profiler, using structurefunctions allows an improvement of temporal resolution, the mitigationof external interference and clutter effects, the determination of thevariances of the horizontal turbulent velocities and the horizontalmomentum flux, and the derivation of operational equations by making asmaller number of less restrictive assumptions. Notwithstanding, thelimitations of the structure function-based method have also been wellrecognized.

To understand the limitations, one should consider the definition andinterpretation of structure functions for the received signals s({rightarrow over (x)}₁,t) and s({right arrow over (x)}₂,t) from two sensors inclose spatial locations {right arrow over (x)}₁ and {right arrow over(x)}₂ at two close times t₁ and t₂. The cross structure function of theorder p is defined as follows:D _(p)({right arrow over (x)},δ{right arrow over (x)},t,τ)=<Δs^(p)({right arrow over (x)},δ{right arrow over (x)},t,τ)>, Δs({rightarrow over (x)},δ{right arrow over (x)},t,τ)=s({right arrow over(x)},t)−s({right arrow over (x)}+δ{right arrow over (x)},t+τ)   (1)Hereinafter: t is time, {right arrow over (x)}={right arrow over (x)}₁,t=t₁, δ{right arrow over (x)}={right arrow over (x)}₂−{right arrow over(x)}₁ and τ=t₂−t₁ are respectively the spatial separation between thesensors and the temporal separation between the signals, and the angularbrackets < > denote ensemble averages. It is important that equation (1)defines only one equation of order p for a pair of sensors. One can seethat D_(p)({right arrow over (x)},δ{right arrow over (x)},t,τ) is thep^(th) order statistical moment of the increment Δs({right arrow over(x)},δ{right arrow over (x)},t,τ); the latter is customarily interpretedas a band-pass filter extracting fluctuations with spatial and temporalscales |δ{right arrow over (x)}| and τ, respectively. However, it haslong been established that this is not the case and that the incrementis, in fact, a multi-band filter. For example, the normalized spectraltransfer function of the auto increment Δs({right arrow over(x)},0,t,τ)=s({right arrow over (x)},t)−s({right arrow over (x)},t+τ) is1−cos(2π f τ) with maxima occurring at multiple frequenciesf=1/(2τ)+k/τ, k=0, 1, 2, . . . . Customarily, only the first band at k=0is taken into account in the interpretation of structure functions,while others of the same intensity and bandwidth are merely ignored. Thenext issue is that a cross structure function is not a rigorousmathematical tool. It follows from equation (1) that the temporalD_(p)({right arrow over (x)},0,t,τ) and spatial D_(p)({right arrow over(x)},δ{right arrow over (x)},t,0) auto structure functions at |τ|→0 and|δ{right arrow over (x)}|→0 are the first-order finite approximations ofthe respective temporal derivatives and spatial derivatives in thedirection δ{right arrow over (x)}. The first order approximation of across derivative at |δ{right arrow over (x)}|→0, |τ|→0 is:

$\begin{matrix}\begin{matrix}{\frac{\partial^{2}{s\left( {\overset{\rightarrow}{x},t} \right)}}{{\partial t}{\partial\overset{\rightarrow}{x}}} \approx {\frac{1}{\tau{{\delta\overset{\rightarrow}{x}}}}\left\lbrack {{s\left( {{\overset{\rightarrow}{x} + {\delta\overset{\rightarrow}{x}}},{t + \tau}} \right)} - {s\left( {{\overset{\rightarrow}{x} + {\delta\overset{\rightarrow}{x}}},t} \right)} - {s\left( {\overset{\rightarrow}{x},{t + \tau}} \right)} + {s\left( {\overset{\rightarrow}{x},t} \right)}} \right\rbrack}} \\{\equiv {{- \frac{1}{\tau{{\delta\overset{\rightarrow}{x}}}}}\left\{ {\left\lbrack {{s\left( {\overset{\rightarrow}{x},t} \right)} - {s\left( {{\overset{\rightarrow}{x} + {\delta\overset{\rightarrow}{x}}},{t + \tau}} \right)}} \right\rbrack - \left\lbrack {{s\left( {\overset{\rightarrow}{x},t} \right)} -} \right.} \right.}} \\\left. {\left. {s\left( {{\overset{\rightarrow}{x} + {\delta\overset{\rightarrow}{x}}},t} \right)} \right\rbrack - \left\lbrack {{s\left( {\overset{\rightarrow}{x},t} \right)} - {s\left( {\overset{\rightarrow}{x},{t + \tau}} \right)}} \right\rbrack} \right\}\end{matrix} & (2)\end{matrix}$One can see from equations (1) and (2) that the increment Δs({rightarrow over (x)},δ{right arrow over (x)},t,τ) corresponds to the firstbracketed term in the derivative while the second and third terms aremerely ignored. Therefore, the cross structure function is a truncatedrepresentation of the cross derivative ∂²s({right arrow over(x)},t)/(∂t∂{right arrow over (x)}).

These theoretical issues lead to serious practical drawbacks when usingstructure function-based data processing methods with monitoringequipment having multiple sensors. For example, in the case of anatmospheric spaced antenna profiler, the major drawbacks are as follows:(1) an inability to retrieve the vertical momentum fluxes, (2) a highsensitivity to white noise, (3) an inability to directly measure thecorrelation between noise from different sensors, and (4) an inabilityto provide more than one equation for each pair of sensors. Thesedrawbacks complicate the operational use of existing data processingmethods and cause degradation in the performance of monitoring equipmentwith multiple sensors.

SUMMARY OF THE INVENTION

It is accordingly an object of the invention to provide a system andmethod for measuring one or more selected characteristics of continuousmedium and/or localized targets in a predetermined volume using multiplesensors, which overcome the above-mentioned disadvantages of the priorart systems and methods of this general type.

With the foregoing and other objects in view, there is provided, asystem for obtaining data indicative of at least one characteristic of acontinuous medium or at least one localized target located within apredetermined volume of space. The system includes a sensorconfiguration and a processing circuit. The sensor configurationincludes a plurality of sensors for acquiring a plurality of signalsfrom the continuous medium or the target. The plurality of sensors havecenters spatially separated from each other in at least one spatialdimension. The system also includes a processing circuit for obtainingthe data indicative of the characteristic or characteristics of thecontinuous medium or the target by calculating a plurality of poweredweighted increments using the plurality of signals acquired by thesensor configuration. The processing circuit then relates the pluralityof powered weighted increments to the characteristic or characteristicsof the continuous medium or the target using a plurality of models. Thedata indicative of characteristic or characteristics is then availableto be output by an output circuit.

In accordance with an added feature of the invention, the plurality ofsensors is configured for concurrently acquiring the plurality ofsignals from the continuous medium or the target.

In accordance with an additional feature of the invention, each one ofthe plurality of sensors is positioned at a predetermined locationinside the predetermined volume of space.

In accordance with another feature of the invention, the plurality ofsensors is positioned outside the predetermined volume of space, and theplurality of signals acquired by the plurality of sensors is generatedby the continuous medium or the target.

In accordance with a further feature of the invention, the plurality ofsensors is positioned outside the predetermined volume of space, and theplurality of signals acquired by the plurality of sensors is caused bythe predetermined radiation that is generated and propagated through thepredetermined volume of space to induce the backscatter from thecontinuous medium or the target.

In accordance with a further added feature of the invention, theprocessing circuit is configured for increasing an amount of informationextractable from the plurality of signals by modifying the plurality ofsignals and subsequently obtaining the data indicative of thecharacteristic of the continuous medium or the target.

In accordance with a further additional feature of the invention, theprocessing circuit is configured for modifying the plurality of signalsby performing at least one modification step selected from a groupconsisting of: converting the plurality of signals from complex signalsto real signals, removing noise from the plurality of signals, removingmean values from the plurality of signals, normalizing each one of theplurality of signals with a standard deviation of the respective one ofthe plurality of signals, and generating virtual sensors usingcombinations of the plurality of signals.

In accordance with another further feature of the invention, theprocessing circuit is configured for calculating the plurality ofpowered weighted increments for one or more specified orders, specifiedpairs of signals from the plurality of the sensors, and specifiedcombinations of weights.

In accordance with yet an added feature of the invention, the processingcircuit is configured for relating the plurality of powered weightedincrements to the characteristic of the medium or target by: fitting theplurality of powered weighted increments to a plurality of predeterminedmodels, estimating a plurality of adjustable parameters in the pluralityof predetermined models, and relating the plurality of adjustableparameters to the characteristic of the continuous medium or the target.

In accordance with yet an additional feature of the invention, at leastone of the plurality of predetermined models is formed as adecomposition into a Taylor series.

In accordance with yet a further feature of the invention, each one ofthe plurality of predetermined models is constructed from at least onemodel selected from a group consisting of an analytically derivedoperational equation formed as a decomposition into a polynomialfunction over a selected parameter, a tabulated function obtained usinga numerical simulation, and a tabulated function obtained using aphysical experiment.

In accordance with yet another added feature of the invention, theprocessing circuit is configured for increasing an accuracy of the dataindicative of the characteristic of the medium or the target byanalyzing multiple estimates of the data indicative of thecharacteristic of the continuous medium or the target.

In accordance with yet another additional feature of the invention, anoutput circuit is provided for outputting the data indicative of thecharacteristic or characteristics of the continuous medium or thetarget.

With the foregoing and other objects in view, there is also provided, amethod for obtaining data indicative of at least one characteristic of acontinuous medium or at least one localized target located within apredetermined volume of space. The method includes steps of: using asensor configuration having a plurality of sensors with centersspatially separated from each other in at least one spatial dimension toacquire a plurality of signals from the continuous medium or the target,and obtaining the data indicative of the characteristic orcharacteristics of the continuous medium or the target by calculating aplurality of powered weighted increments using the plurality of signalsacquired by the plurality of sensors and by relating the plurality ofpowered weighted increments to the characteristic or characteristics ofthe medium or the target using a plurality of models.

In accordance with an added mode of the invention, the method includesconcurrently acquiring the plurality of signals from the continuousmedium or the target with the plurality of sensors.

In accordance with an additional mode of the invention, the methodincludes positioning each one of the plurality of sensors at apredetermined location inside the predetermined volume of space.

In accordance with another mode of the invention, the method includespositioning each one of the plurality of sensors at a predeterminedlocation outside the predetermined volume of space. This is typicallyperformed when the plurality of signals acquired by the plurality ofsensors are generated by the continuous medium or the target.

In accordance with a further mode of the invention, the method includes:positioning each one of the plurality of sensors at a predeterminedlocation outside the predetermined volume of space, and generating andpropagating the predetermined radiation through the predetermined volumeof space to induce the backscatter from the continuous medium or thetarget in a manner enabling the plurality of signals to be acquired bythe plurality of sensors.

In accordance with a further added mode of the invention, the methodincludes: increasing an amount of information extractable from theplurality of signals by modifying the plurality of signals, andsubsequently performing the step of obtaining the data indicative of thecharacteristic of the continuous medium or the target.

In accordance with a further additional mode of the invention, themethod includes increasing an amount of information extractable from theplurality of signals by performing a modification step selected from agroup consisting of: converting the plurality of signals from complexsignals to real signals, removing noise from the plurality of signals,removing mean values from the plurality of signals, normalizing each oneof the plurality of signals with a standard deviation of the respectiveone of the plurality of signals, and generating virtual sensors usingcombinations of the plurality of signals. Subsequently, the step ofobtaining the data indicative of the characteristic of the continuousmedium or the target is performed.

In accordance with yet an added mode of the invention, the methodincludes calculating the plurality of powered weighted increments forone or more specified orders, specified pairs of signals from theplurality of the sensors, and specified combinations of weights.

In accordance with yet an additional mode of the invention, the methodincludes relating the plurality of powered weighted increments to thecharacteristic of the medium or target by: fitting the plurality ofpowered weighted increments to a plurality of predetermined models,estimating a plurality of adjustable parameters in the plurality ofpredetermined models, and relating the plurality of adjustableparameters to the characteristic of the continuous medium or the target.

In accordance with yet another mode of the invention, the methodincludes forming at least one of the plurality of predetermined modelsas a decomposition into a Taylor series.

In accordance with yet a further mode of the invention, the methodincludes constructing each one of the plurality of predetermined modelsfrom at least one model selected from a group consisting of: ananalytically derived operational equation formed as a decomposition intoa polynomial function over a selected parameter, a tabulated functionobtained using a numerical simulation, and a tabulated function obtainedusing a physical experiment.

In accordance with yet a further added mode of the invention, the methodincludes increasing an accuracy of the data indicative of thecharacteristic of the continuous medium or the target by analyzingmultiple estimates of the data indicative of the characteristic of thecontinuous medium or the target.

In accordance with yet a concomitant mode of the invention, the methodincludes outputting the data indicative of the characteristic of thecontinuous medium or the target.

The present invention is based on the development of powered weightedincrements, which is a new mathematical tool for data processing. Aplurality of powered weighted increments can be used with a plurality ofreceived signals in order to obtain data indicative of selectedcharacteristics of the medium or the target. By processing a pluralityof powered weighted increments obtained using a plurality of receivedsignals in the manner described herein, the present invention overcomesmany of the problems found in existing monitoring equipment. Anon-exhaustive list of such problems that have now been solved with thepresent invention includes: the ability to retrieve importantcharacteristics of the monitored object, such as, for example, thevertical momentum fluxes in the case of an atmospheric profiler, theability to directly measure the correlation between noise from differentsensors, and the ability to provide more than one equation for each pairof sensors.

A key feature of the present invention is using powered weightedincrements to determine selected characteristics of the object frommultiple received signals, rather than using correlation functions,spectra, or structure functions of the signals. The powered weightedincrements enable the use of an unlimited number of equations withvariable weights for a plurality of signals, which preferably come froma plurality of physical sensors, in order to obtain the most efficientand reliable estimation of each selected characteristic of the monitoredobject. The present invention allows the estimation of selectedcharacteristics of the medium or the target with a higher accuracy andan improved temporal and/or spatial resolution compared to prior artsystems and methods. When compared to the prior art, the presentinvention functions more reliably in adverse conditions, allows thedetection and identification of specified targets that could not bedetected and/or identified otherwise, allows the retrieval ofcharacteristics of the monitored medium that could not be retrievedotherwise, and provides measurement error for each estimatedcharacteristic of the monitored object. The powered weighted incrementsare presented to mathematical models with adjustable parameters andthese parameters are related to the selected characteristics of themonitored object that are to be determined. The models can be analyticalexpressions, tabulated results of numerical simulations or experiments,and the like, and the adjustable parameters can be estimated usingstandard, well-known fitting methods. The present invention enablesconstructing models for any type and configuration of detection andmonitoring equipment, any operational mode of the detection andmonitoring equipment, and any of a number of selected characteristics ofthe object. The preferred mathematical models are analytical operationalequations in the form of decompositions into polynomial functions over asufficiently small temporal and/or spatial separation, where theadjustable parameters are the coefficients in the decompositions. Whenpreferred conditions for using the present invention are satisfied, thepreferred models can be constructed by making a smaller number of lessrestrictive assumptions than would be possible with prior art methods.The present invention can be applied to detection and monitoringequipment receiving multiple signals related to known characteristics ofthe target or medium independent of the physical nature of the signalsand the configuration of the data acquisition device. Such detection andmonitoring equipment can deploy in-situ sensors and/or passive or activeremote sensors that can be mounted on a fixed or moving platform.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram showing steps that may be performed in apreferred embodiment of the system and method for measuring selectedcharacteristics of continuous medium and/or localized targets;

FIG. 2A is a schematic illustration of a single, three-paneltransmitting antenna that can be used in an atmospheric profiling radarbased system;

FIG. 2B is a schematic illustration of a plurality of identicalspatially separated physical sensors for estimating selectedcharacteristics of the atmosphere in a predetermined region of space;

FIG. 3 is a schematic illustration showing a transmitter illuminating avolume in order to receive signals in a predetermined sub-volume of theilluminated volume;

FIGS. 4A, 4B, and 4C are schematic illustrations showing a configurationof virtual sensors that are generated using actual signals obtained fromthe physical sensors depicted in FIG. 2B; and

FIG. 5 is a block diagram showing a preferred embodiment of a system formeasuring selected characteristics of continuous medium and/or localizedtargets.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In describing the present invention we will begin with an introductionof the new concept of powered weighted increments and a basic overviewof a preferred embodiment of the inventive method. Later on, a specificexample is provided solely for the purpose of illustrating some of thefeatures of the invention. The invention should not be construed asbeing limited to being applied in the manner shown in the example or inthe preferred embodiment. The invention relates to a method as well as asystem for implementing the method. The invention provides an importantadvance in multi-sensor signal processing—the key feature of which isthe use of a plurality of powered weighted increments to determine thecharacteristics of the object from a plurality of signals received froma plurality of sensors. Powered weighted increments provide substantialadvantages over conventional correlation function, spectrum or structurefunction-based methods. Powered weighted increments Φ_(p), which are anew mathematical tool for processing signals from a pair of sensors (thepair can include two physical sensors, two virtual sensors as well asone physical and one virtual sensor) in spatially separated locations{right arrow over (x)}_(s1) and {right arrow over (x)}_(s2) attemporarily separated instances t₁ and t₂, are defined as:Φ_(p)({right arrow over (x)}_(s1),{right arrow over (x)}_(s2) ,t ₁ ,t₂,ω_(x),ω_(τ))=<{[s({right arrow over (x)}_(s1) ,t ₁)−s({right arrowover (x)}_(s2) ,t ₂)]−ω_(x) [s({right arrow over (x)}_(s1) ,t₁)−s({right arrow over (x)}_(s2) ,t ₁)]−ω_(τ) [s({right arrow over(x)}_(s1) ,t ₁)−s({right arrow over (x)}_(s1) ,t ₂)]}^(p)>  (3)The angular brackets < > in the definition denote the ensemble averages,the temporal averages over a specified time interval, or the spatialaverages over a specified spatial domain. The weights −∞<ω_(x), ω_(τ)<∞are free parameters that may have arbitrary real values. It should benoted that both weights ω_(x) and ω_(τ) may take on zero values at thesame time for calculating some of the powered weighted increments, butthey both cannot always be zero when calculating a plurality of poweredweighted increments for determining a particular characteristic or setof characteristics of a monitored object. The powered weightedincrements (3) evaluated as |{right arrow over (x)}_(s2)−{right arrowover (x)}_(s1)|→0, |t₂−t₁|→0, and with ω_(x)=ω_(τ)=1 exactly approximatethe cross derivative (2). It is important that the power p in definition(3) be applied before averaging. Therefore, the function Φ_(p) is not alinear combination of the individual increments in the square brackets.Using equation (3) one may obtain an unlimited number of equations for apair of signals. The system and method for measuring characteristics ofcontinuous medium and/or localized targets is therefore based onobtaining a plurality of powered weighted increments from a plurality ofreceived signals—a new mathematical tool for data processing.

The present invention enables the measurement of characteristics ofcontinuous medium and/or localized targets using a sensor configurationthat includes a plurality of physical sensors. The present inventionovercomes the above-noted problems including the inability to retrievemore characteristics of the monitored object such as the verticalmomentum fluxes for the case of an atmospheric profiler, the inabilityto directly measure the correlation between noises from differentsensors, and the inability to provide more than one equation for eachpair of sensors. The invention enables one to obtain efficient andreliable estimates for each selected characteristic of the monitoredobject by choosing an appropriate number of equations in the form ofequation (3) in which each such equation is configured with the optimalcombination of weights ω_(x) and ω_(τ) and power p for eachcharacteristic that is to be determined. The present system and methodalso improves the temporal and/or spatial resolution, decreases thesensitivity to white noise and further mitigates or removescontamination of the results by external interferences and clutter.

The present system and method enable one to determine a selected set ofcharacteristics of a predetermined object, where the number and locationof sensors in this system depend on the characteristics of the objectthat will be measured. Notably, however, at least two spatiallyseparated sensors must be deployed. The present invention is capable ofdetermining various characteristics of a monitored object, including butnot limited to the: size, shape, visibility, speed and direction of theobject's motion, rates of changes of the above characteristics, and thelike.

Referring now to the figures of the drawing in detail and first,particularly, to FIG. 1 thereof, there is shown a block diagramillustrating the steps involved in a generalized exemplary embodiment ofthe method for measuring selected characteristics of continuous mediumand/or localized targets. The system would implement these method steps.It should be noted, however, that not all shown steps must necessarilybe performed. It is conceivable that in some cases performing only steps101, 103, 104, and 105 would be sufficient and acceptable and it isintended that the invention not be necessarily limited beyond thesesteps. The characteristics of the continuous medium and/or localizedtargets would then be available for display, archiving, or transfer toanother device as is indicated in step 107.

As shown in FIG. 1, at step 101 the system collects received signalsfrom the sensor configuration. The system configuration and operationalparameters and the data processing parameters may be specified at thisstep or may have been previously specified.

The actual signals that have been received or obtained from each sensorin the sensor configuration may be modified at step 102 to ensureefficient estimation of the selected characteristics of the medium orthe target. The modification of the actual signals may include any of anumber of known operations, such as, for example, combining signals froma plurality of physical sensors to generate virtual sensors, normalizingthe signals, and other modification processes which will become apparentto one of ordinary skill in the art after considering the instantdisclosure. Such modification steps may be performed by electroniccircuitry constructed as part of the sensor configuration or may beperformed by the processing circuit.

A plurality of powered weighted increments with specified orders andspecified weights are then calculated at step 103 for each selected pairof signals obtained from the sensor configuration.

In step 104, the powered weighted increments that were calculated instep 103 are fit to predetermined models and adjustable parameters inthe models are estimated using standard fitting techniques. It ispreferable that such models be analytically derived operationalequations in the form of decompositions into polynomial functions over asmall parameter; it is also notable however, that such models can alsobe, for example, tabulated functions obtained using numericalsimulations or physical experiments.

In step 105, the adjustable parameters are related to the selectedcharacteristics of the continuous medium or localized target. Specificmodels for relating the adjustable parameters to the selectedcharacteristics of the monitored object depend on: the type,configuration, and operational mode of the monitoring equipment, theunderlying theory and techniques that are chosen for constructing themodels of the object, and the characteristics of the object that are tobe determined. The adjustable parameters are preferably related to theselected characteristics using analytical expressions, however,discretely tabulated curves may also be used for this purpose.

Multiple estimates for each selected characteristic of the monitoredobject may then be analyzed at step 106 to provide the best estimate foreach characteristic, to obtain the accuracy for the best estimate, and,if this is required, to obtain a measure for the reliability of the bestestimates. This processing step can include a statistical analysis ofall available estimates for the selected characteristics of the object,a joint statistical analysis of the selected characteristics at theanalyzed time interval with those from a previous time interval(s), andother analysis techniques which will become apparent to one of ordinaryskill in the art after considering the instant disclosure.

The best and/or all multiple estimates for the selected characteristicsof the monitored object can be displayed at step 107 in any userspecified format. The set of characteristics can also be transferred toa remote location and/or archived for future use and/or reference.

Sensor Configuration

FIGS. 2A, 2B, and 3 are provided to illustrate an example of atransmitter and sensor configuration that can be used to implement thepresent system for measuring characteristics of continuous medium and/orlocalized targets. The example is intended to demonstrate the mainfeatures and advantages of the present invention compared with multiplesensor measurement systems implementing prior art data processingmethods. This example is provided for illustrative purposes only and isnot intended to limit the areas of application or the scope of thepresent invention.

The illustrative example is an atmospheric profiling radar based systemthat uses the single three-panel transmitting antenna T shown in FIG. 2Aand the plurality of identical spatially separated physical sensors A₁,A₂, and A₃ shown in FIG. 2B to estimate selected characteristics of theatmosphere in a predetermined region of space V. The characteristic sizeof the transmitting antenna T is denoted by D in FIG. 2A and thecharacteristic size of the receiving antennas or sensors A₁, A₂, and A₃is denoted by D_(A) in FIG. 2B. FIG. 3 shows that in this system, thetransmitting antenna T transmits a focused beam of radio frequencyenergy to illuminate the volume of space that contains the predeterminedvolume V. The focused beam is formed from a series of pulses and eachpulse has a pulse origination time at which the pulse is emanated fromthe transmitting antenna T. The transmitted pulses of radio frequencyenergy are scattered by the refractive index irregularities in theatmosphere. Following standard terminology, these irregularities arereferred to as scatterers independent of their nature and type ofscattering. A portion of the scattered waves reaches the sensors A₁, A₂,and A₃ (each panel in the transmitting antenna T operates as a separatesensor in this system) and induces signals in the independent receivingchannels. The signals are collected in the selected range gates and areprocessed to estimate the characteristics of the atmosphere in eachpredetermined range gate or predetermined volume V with a center-rangeheight R above the ground as shown in FIG. 3. An adaptive Cartesiancoordinate system {right arrow over (x)}={x, y, z} with a verticallydirected z axis and x and y axes in a horizontal plane is usedhereinafter, in which symbols in brackets { } denote the Cartesiancomponents of a vector. For each predetermined volume V, the adaptiveorigin of the coordinate system is placed at a height R above the centerof the transmitting antenna T. The bullets in FIGS. 2A and 2B indicatethe antenna phase centers with the coordinates {right arrow over(x)}_(T)={0,0,−R} for the transmitting antenna T shown in FIG. 2A and{right arrow over (x)}_(Am)={x_(Am),y_(Am),−R} for the receiving sensorsA_(m), m=1, 2, and 3 shown in FIG. 2B.

The current induced in the internal resistance R_(int) of thematched-filter sensor A_(m) (m=1, 2, or 3 in this system) can bepresented in standard complex form as:

$\begin{matrix}{{{I_{Am}(t)} + {{jQ}_{Am}(t)}} = {C{\sum\limits_{i = 1}^{N}{\frac{{W_{T}\left( {\overset{\rightarrow}{x}}_{i} \right)}{g_{T}^{1/2}\left( {\overset{\rightarrow}{x}}_{i} \right)}{g_{Am}^{1/2}\left( {\overset{\rightarrow}{x}}_{i} \right)}\Delta\;{n\left( {{\overset{\rightarrow}{x}}_{i},t} \right)}}{{r_{T}\left( {\overset{\rightarrow}{x}}_{i} \right)}{r_{Am}\left( {\overset{\rightarrow}{x}}_{i} \right)}}\exp\left\{ {{- j}{\frac{2\pi}{\lambda}\left\lbrack {{r_{T}\left( {\overset{\rightarrow}{x}}_{i} \right)} + {r_{Am}\left( {\overset{\rightarrow}{x}}_{i} \right)}} \right\rbrack}} \right\}}}}} & (4)\end{matrix}$Here I_(Am)(t) and Q_(Am)(t) are the in-phase and quadrature componentsof the induced signal; the subscript i identifies the i^(th) scatterer;N is the number of scatterers; Δn({right arrow over (x)}_(i),t) is themagnitude of fluctuation in the refractive index field (the refractiveindex irregularity) produced by a scatterer in the location {right arrowover (x)}_(i)(t)={x_(i)(t),y_(i)(t),z_(i)(t)} at instant t; W_(T)({rightarrow over (x)}_(i)), g_(T)({right arrow over (x)}_(i)), andg_(Am)({right arrow over (x)}_(i)) are the range weighting function, thetransmitting antenna gain function, and the sensor's field of view gainfunction, respectively; r_(T)({right arrow over (x)}_(i)) andr_(Am)({right arrow over (x)}_(i)) are distances to the scatterer fromthe phase centers of the transmitting antenna T and sensor A_(m); λ isthe wavelength of the transmitted signal; and j=√{square root over(−1)}. The constant C is given by C=√{square root over(P_(T)/(2R_(int)))}/λ where P_(T) is the average power of thetransmitted pulse. Equation (4) is valid for the far zone r_(T)≧(2D²/λ)and r_(Ak)≧(2D_(A) ²/λ). To satisfy the far zone requirements,sufficiently large ranges are considered below, such as:R>>D,D_(A),σ_(r)   (5)Hereinafter σ_(r) denotes the width of the range weighting function inEq. (12) below.

At step 101 in FIG. 1, the present system for measuring characteristicsof continuous medium and/or localized targets acquires the receivedsignals I_(Am)(t) and Q_(Am)(t) from all deployed physical sensorsA_(m), m=1,2, and 3 during a predetermined period of measurementT_(meas) starting at the predetermined instant t₀. Each sensor providessignals for a predetermined set of the center ranges R_(l), l=1, 2, . .. , L corresponding to different volumes V_(l). For a pulse wave radar,I_(Am)(t) and Q_(Am)(t) are discrete time series with the sampling timeinterval δt=NCI/PRF where PRF and NCI denote the transmitter pulserepetition frequency and the number of coherent integrations. Allrelevant operational parameters of the system such as t₀, T_(meas),R_(l), δt, and others are specified at step 101 in FIG. 1.

The technical parameters of the system are also specified at step 101.These parameters include the transmitter wavelength λ and its apertureD, the number of sensors, their aperture D_(A) and coordinates of theircenters {right arrow over (x)}_(Am) (m=1, 2, and 3 in this system), andothers.

The data processing parameters are specified at step 101 in FIG. 1 aswell. In this system, the ensemble averages in the definition (3) andother theoretical equations are substituted by temporal averages overthe predetermined time interval T_(av); the latter is specified at thisstep. The pairs of sensors to be analyzed, the orders p of the poweredweighted increments, the weights ω_(x), ω_(τ) in Eq. (3), and otherparameters are specified.

Modification of Signals from each Sensor

The actual received signals from all deployed physical sensors aremodified at step 102 in FIG. 1 to ensure the most efficient extractionof useful information from multiple signals. Modification may include(although is not limited by) the following operations.

(a) Powered weighted increments are the most efficient analysis toolwhen applied to real signals. The complex signal from the in-phase andquadrature synchronous detector given by Eq. (4) can be converted intothe real instantaneous signal power according to the following equation:S({right arrow over (x)} _(Am) ,t)=I _(Am) ²(t)+Q _(Am) ²(t), m=1, 2, 3  (6)The pure received power S({right arrow over (x)}_(Am),t) from scatterersin the predetermined volume V in FIG. 3, with no contribution fromnoise, clutter, or other contaminants, is referred to below as a purereceived signal from range R from sensor A_(m) with a center {rightarrow over (x)}_(Am).

(b) In practical measurements, the received signal contains both purereturn from a monitored object S({right arrow over (x)}_(Am),t) andnoise n({right arrow over (x)}_(Am),t), therefore the practical receivedsignal from the sensor A_(m) can be presented as{tilde over (s)}({right arrow over (x)} _(Am) ,t)=S({right arrow over(x)} _(Am) ,t)+n({right arrow over (x)} _(Am) ,t), m=1, 2, 3   (7)At step 102 one can apply a high-pass, low-pass, and/or band-pass filterto partly remove noise n({right arrow over (x)}_(Am),t) and increase thesignal-to-noise ratio.

(c) To eliminate possible offsets, one can remove the mean values<{tilde over (s)}({right arrow over (x)}_(Am),t)>, m=1, 2, 3, from thesignals.

(d) To simplify computations, one can normalize the signals with theirstandard deviations √{square root over (<└{tilde over (s)}({right arrowover (x)}_(Am),t)−<{tilde over (s)}({right arrow over(x)}_(Am),t)>┘²>)}, m=1, 2, 3.

(e) One can increase the number of sensors in the system without changesin its hardware configuration by generating virtual sensors usingcombinations of the actual received signals. For example, one cangenerate received signals from three virtual sensors A₄, A₅, and A₆ inthis system using the following equations:{tilde over (s)}({right arrow over (x)} _(A4) ,t)=[I _(A1)(t)+I_(A2)(t)]² +[Q _(A1)(t)+Q _(A2)(t)]²{tilde over (s)}({right arrow over (x)} _(A4) ,t)=[I _(A2)(t)+I_(A3)(t)]² +[Q _(A2)(t)+Q _(A3)(t)]²{tilde over (s)}({right arrow over (x)} _(A4) ,t)=[I _(A1)(t)+I_(A3)(t)]² +[Q _(A1)(t)+Q _(A3)(t)]²  (8)These virtual sensors with the phase centers {right arrow over(x)}_(A4), {right arrow over (x)}_(A5), and {right arrow over (x)}_(A6)are illustrated in FIGS. 4A, 4B, and 4C. The signal power is anon-linear function of I_(Am)(t) and Q_(Am)(t), hence the signals {tildeover (s)}({right arrow over (x)}_(A4),t), {tilde over (s)}({right arrowover (x)}_(A5),t), and {tilde over (s)}({right arrow over (x)}_(A6),t)given by equations (8) are non-linear combinations of the actual signalsfrom sensors A₁, A₂, and A₃. The virtual sensors A₄, A₅, A₆ have alarger aperture and, therefore provide a higher signal-to-noise ratiothan is provided by the actual sensors A₁, A₂, and A₃. In addition, thecombined signals are more strongly correlated among themselves than arethe actual signals. Signals from the virtual sensors A₄, A₅, A₆ aretreated in the same way as the signals from the actual sensors A₁, A₂,and A₃. For example, the above operations (b), (c) and (d) can also beapplied to the combined signals (8).Calculating and Analyzing the Powered Weighted Increments

The objective step 103 is to calculate powered weighted increments forall pairs of sensors, gate heights, weights, and averaging timeintervals that are specified in step 101. For each gate range R andaveraging time interval specified at step 101, one calculates a poweredweighted increment for each specified power p, specified pair of sensorsA_(m1) and A_(m2), and specified combination of weights ω_(x) and ω_(τ).The powered weighted increment Φ_(p) is calculated as follows:{tilde over (Φ)}_(p)({right arrow over (x)} _(Am1) ,{right arrow over(x)} _(Am2) ,t,τ,ω _(x),ω_(τ))=<{[{tilde over (s)}({right arrow over(x)} _(Am1) ,t)−{tilde over (s)}({right arrow over (x)} _(Am2),t+τ)]−ω_(x) [{tilde over (s)}({right arrow over (x)} _(Am1) ,t)−{tildeover (s)}({right arrow over (x)} _(Am2) ,t)]−ω_(τ) [{tilde over(s)}({right arrow over (x)} _(Am1) ,t)−{tilde over (s)}({right arrowover (x)} _(Am1) ,t+τ)]}^(p)>  (9)For each selected combination of {right arrow over (x)}_(Am1), {rightarrow over (x)}_(Am2), t, ω_(x), and ω_(τ), the calculated increment{tilde over (Φ)}({right arrow over (x)}_(Am1),{right arrow over(x)}_(Am2),t,τ,ω_(x),ω_(τ)) provides a confined set of observations atthe temporal separations τ such as ±δt, ±2δt . . . specified at step101.

As previously discussed, in step 104, the powered weighted incrementsthat were calculated in step 103 are fit to predetermined models andadjustable parameters in the models are estimated using standard fittingtechniques. In a preferred embodiment of the invention, the poweredweighted increments are fit to mathematical models in the analyticalform of polynomial decompositions, and adjustable parameters(coefficients in the decompositions) are estimated. Implementing thepreferred embodiment of the invention requires the deployed equipment tobe operable in a manner so as to provide a sufficiently small temporalseparation τ between signals and/or spatial separation |δ{right arrowover (x)}_(m2,m1)|=|{right arrow over (x)}_(Am2)−{right arrow over(x)}_(Am1)| between the sensors. One can always use a sufficiently highpulse repetition frequency and a sufficiently small number of coherentintegrations to get sufficiently small temporal separations τ=±δt, ±2δt. . . , as will be seen later in the description of an example in theform of an atmospheric profiling radar based system. When the temporalrequirement is satisfied, each powered weighted increment at very smallτ→0 can be decomposed in the Taylor series as follows:{tilde over (Φ)}_(p)({right arrow over (x)} _(Am1) ,{right arrow over(x)} _(Am2) ,t,τ,ω _(x),ω_(τ))={tilde over (d)} _(p,0)({right arrow over(x)} _(Am1) ,{right arrow over (x)} _(Am2) ,t,ω _(x),ω_(τ))+τ{tilde over(d)} _(p,1)({right arrow over (x)} _(Am1) ,{right arrow over (x)} _(Am2),t,ω_(x),ω_(τ))+τ² {tilde over (d)} _(p,2)({right arrow over (x)} _(Am1),{right arrow over (x)} _(Am2) ,t,ω _(x),ω_(τ))+O(τ³)  (10)Decomposition (10) is an example of a preferred model that can be usedwith the powered weighted increments, where the coefficients {tilde over(d)}_(p,0), {tilde over (d)}_(p,1), and {tilde over (d)}_(p,2) are theadjustable parameters. In performing step 104 according to the preferredembodiment, the coefficients can be estimated for each calculatedincrement {tilde over (Φ)}({right arrow over (x)}_(Am1), {right arrowover (x)}_(Am2),t,τ,ω_(x),ω_(τ)) as the best-fit parameters using any ofthe standard fitting procedures, for example, the least squares as amaximum likelihood estimator. In most practical cases the coefficients{tilde over (d)}_(p,0), {tilde over (d)}_(p,1), and {tilde over(d)}_(p,2) in Eq. (10) can be analytically related to the selectedcharacteristics of the monitored object as will be explained indescribing the preferred manner of implementing step 105. An example ofsuch analytical relations, which are referred to below as operationalequations, is given in the next section for the case of an atmosphericprofiling radar based system.

When the spatial requirement is satisfied, each powered weightedincrement at very small |δ{right arrow over (x)}_(m2,m1)|→0 can bedecomposed in the Taylor series as follows:{tilde over (Φ)}_(p)({right arrow over (x)} _(Am1) ,{right arrow over(x)} _(Am2) ,t,τ,ω _(x),ω_(τ))={tilde over (d)} _(p,x0)({right arrowover (x)} _(Am1) ,t,τ,ω _(x),ω_(τ))+δ{right arrow over (x)} _(m2,m1){tilde over (d)} _(p,x1)({right arrow over (x)} _(Am1) ,t,τ,ω_(x),ω_(τ))+δ{right arrow over (x)}_(m2,m1) ² {tilde over (d)}_(p,x2)({right arrow over (x)} _(Am1) ,t,τ,ω _(x),ω_(τ))+O(|δ{rightarrow over (x)} _(m2,m1)|³)   (10a)Decomposition (10a) is another example of the preferred model for thepowered weighted increments where the coefficients {tilde over(d)}_(p,x0), {tilde over (d)}_(p,x1), and {tilde over (d)}_(p,x2) arethe adjustable parameters. At this step 104, the coefficients can beestimated for each calculated increment {tilde over (Φ)}({right arrowover (x)}_(Am1),{right arrow over (x)}_(Am2),t,τ,ω_(x),ω_(τ)) as thebest-fit parameters using any of the standard fitting procedures. Inmost practical cases the coefficients {tilde over (d)}_(p,x0), {tildeover (d)}_(p,x1), and {tilde over (d)}_(p,x2) in Eq. (10a) can beanalytically related to the selected characteristics of the monitoredobject.Estimating the Selected Characteristics of the Monitored Object

The objective of step 105 is to estimate all of the selectedcharacteristics of the monitored object by relating the adjustableparameters that were estimated in step 104 to the selectedcharacteristics of the monitored object. In the present system andmethod, specific equations for relating the adjustable parameters to thecharacteristics of the object depend on: the type, configuration, andoperational mode of the monitoring equipment, the underlying theory andtechniques that are chosen for constructing the models of the monitoredobject, and the characteristics of the object that are to be determined.For the particular case of an atmospheric profiling radar based systemshown in FIGS. 2, 3, and 4, the models are decompositions (10) and theadjustable parameters are coefficients {tilde over (d)}_(p,0), {tildeover (d)}_(p,1), and {tilde over (d)}_(p,2). Operational equations forobtaining the selected characteristics of the atmosphere in thepredetermined volume of space from the coefficients {tilde over(d)}_(p,0), {tilde over (d)}_(p,1), and {tilde over (d)}_(p,2) that wereestimated with the decomposition (10) are derived and discussed in thefollowing portions of this section.

The Powered Weighted Increments for a Pair of Sensors

To get the expression for the instantaneous power of the pure receivedsignal from sensor A_(m) at instant t, one can combine equations (4) and(6) as follows:

$\begin{matrix}{{{S\left( {{\overset{\rightarrow}{x}}_{Am},t} \right)} = {C^{2}{\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{{A\left( {\overset{\rightarrow}{x}}_{i} \right)}{A\left( {\overset{\rightarrow}{x}}_{k} \right)}\Delta\;{n\left( {{\overset{\rightarrow}{x}}_{i},t} \right)}\Delta\;{n\left( {{\overset{\rightarrow}{x}}_{k},t} \right)}{\cos\left\lbrack {{\phi\left( {\overset{\rightarrow}{x}}_{i} \right)} - {\phi\left( {\overset{\rightarrow}{x}}_{k} \right)}} \right\rbrack}}}}}}{{{A\left( {\overset{\rightarrow}{x}}_{i} \right)} = \frac{{W_{T}\left( {\overset{\rightarrow}{x}}_{i} \right)}{g_{T}^{1/2}\left( {\overset{\rightarrow}{x}}_{i} \right)}{g_{Am}^{1/2}\left( {\overset{\rightarrow}{x}}_{i} \right)}}{{r_{T}\left( {\overset{\rightarrow}{x}}_{i} \right)}{r_{Am}\left( {\overset{\rightarrow}{x}}_{i} \right)}}},{{\phi\left( {\overset{\rightarrow}{x}}_{i} \right)} = {\frac{2\pi}{\lambda}\left\lbrack {{r_{T}\left( {\overset{\rightarrow}{x}}_{i} \right)} + {r_{Am}\left( {\overset{\rightarrow}{x}}_{i} \right)}} \right\rbrack}}}} & (11)\end{matrix}$Hereinafter m=1, 2, . . . , 6, which includes both the physical sensorsA₁, A₂, A₃ shown in FIG. 2B and the virtual sensors A₄, A₅, A₆ shown inFIG. 4. An argument t is omitted whenever it is not confusing, forexample, {right arrow over (x)}_(i)(t) is denoted as {right arrow over(x)}_(i) in equations (4) and (11).

Let us consider any pair of sensors A_(m1) and A_(m2) at (m1≠m2)=1, 2, .. . , 6 and specify Eq. (11) for signals from these sensors. Withoutlosing generality, one can consider the x axis of the adaptive Cartesiancoordinate system in FIG. 3 to be directed along the baseline δ{rightarrow over (x)}_(m2,m1)={right arrow over (x)}_(Am2)−{right arrow over(x)}_(Am1); in this case the separation between the phase centers of thesensors is δ{right arrow over (x)}_(m2,m1)={δx_(m2,m1),0,0}. The rangeweighting function, the transmitting antenna gain function, and thesensor's field of view gain function can be approximated in the chosencoordinate system by the standard expressions:

$\begin{matrix}{{{W\left( {\overset{\rightarrow}{x}}_{i} \right)} = {\exp\left( {- \frac{z_{i}^{2}}{4\sigma_{r}^{2}}} \right)}},{{g_{T}^{1/2}\left( {\overset{\rightarrow}{x}}_{i} \right)} = {\exp\left( {- \frac{x_{i}^{2} + y_{i}^{2}}{4\sigma^{2}}} \right)}},{{g_{Am}^{1/2}\left( \overset{->}{x_{i}} \right)} = {\exp\left( {- \frac{\left( {x_{i} - x_{Am}} \right)^{2} + \left( {y_{i} - y_{Am}} \right)^{2}}{4\sigma_{A}^{2}}} \right)}},{i = 1},2,\ldots\mspace{11mu},N} & (12)\end{matrix}$The range width σ_(r) depends on the width of the transmitted pulse. Thetransmitter beam width σ and the sensor's field of view width σ_(A) canbe approximated by the standard expressions:σ≈γλR/D, σ _(A) ≈γλR/D _(A)   (13)Here γ, γ_(A) are constant for a given radar; in general γ≠γ_(A). Itfollows from inequalities (5) and Eq. (13) that for all i=1, 2, . . . ,N:r _(T)({right arrow over (x)} _(i))r _(Am)({right arrow over (x)}_(i))≈R ² ,|{right arrow over (x)} _(Am) −{right arrow over (x)} _(T)|/σ,|δ{right arrow over (x)} _(m2,m1)|/σ<<1   (14)One can combine equations (5) and (11)-(14) to present a signal S({rightarrow over (x)}_(Am1),t) as:

$\begin{matrix}{{S\left( {{\overset{\rightarrow}{x}}_{{Am}\; 1},t} \right)} = {C_{S}{\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\Delta\; n_{i}\Delta\; n_{k}B_{i}B_{k}\mspace{11mu}{\cos\left( \phi_{ik} \right)}}}}}} & (15)\end{matrix}$Here Δn_(i)=Δn({right arrow over (x)}_(Am1),{right arrow over(x)}_(i),t) is a reflectivity of a scatterer in location {right arrowover (x)}_(i)(t) at the instant t for the sensor A_(m1) with center{right arrow over (x)}_(Am1); the insignificant small terms areneglected hereinafter, and

$\begin{matrix}{{{B_{i} = {\exp\left( {{{- \frac{\alpha^{2}}{4}}\frac{x_{i}^{2}}{\sigma^{2}}} - {\frac{\alpha^{2}}{4}\frac{y_{i}^{2}}{\sigma^{2}}} - {\frac{1}{4}\frac{z_{i}^{2}}{\sigma_{r}^{2}}}} \right)}},{\alpha^{2} = {1 + \frac{\sigma^{2}}{\sigma_{A}^{2}}}}}{{\phi_{ik} = {\phi_{i} - \phi_{k}}},{\phi_{i} = {{\frac{2{\pi\gamma\sigma}}{D}\frac{x_{i}^{2} + y_{i}^{2}}{\sigma^{2}}} + {\frac{2{\pi\sigma}_{r}}{\lambda}\frac{z_{i}}{\sigma_{r}}}}},{C_{s} = \frac{C^{2}}{R^{4}}}}} & (16)\end{matrix}$The magnitudes of α are different for the physical sensors A₁, A₂, A₃and virtual sensors A₄, A₅, A₆; practical implications of the differenceare discussed below. Let us denote the instantaneous velocity of i^(th)scatterer at instant t by {right arrow over (W)}({right arrow over(x)}_(i),t)={U({right arrow over (x)}_(i),t), V({right arrow over(x)}_(i),t), W({right arrow over (x)}_(i),t)}, and make the keyassumptions for deriving operational equations for the atmosphericprofiling radar based system.

Assumption 1: The characteristics of scatterers Δn_(i), x_(i)(t),y_(i)(t), z_(i)(t), U({right arrow over (x)}_(i),t), V({right arrow over(x)}_(i),t), W({right arrow over (x)}_(i),t) at i=1, 2, . . . , N arethe locally statistically stationary random processes.

It follows from this assumption that the ensemble averages of thesecharacteristics are independent of time t and the random vector {rightarrow over (W)}({right arrow over (x)}_(i),t) can be presented as:{right arrow over (W)}({right arrow over (x)} _(i) ,t)=<{right arrowover (W)} _(i) >+{right arrow over (w)} _(i)(t), <{right arrow over (w)}_(i)>=0   (17)Hereinafter <{right arrow over (W)}_(i)>=<{right arrow over (W)}({rightarrow over (x)}_(i),t)> and {right arrow over (w)}_(i)(t)={u_(i)(t),v_(i)(t), w_(i)(t)} describe the mean and turbulent motion of the i^(th)scatterer. Neglecting the small terms O(τ²) at τ→0, one can define thelocation of the i^(th) scatterer at instant t+τ as:{right arrow over (x)} _(i)(t+τ)={right arrow over (x)} _(i)(t)+τ{rightarrow over (W)} _(i)(t)   (18)Neglecting the terms O(|δ{right arrow over (x)}_(m2,m1)|²,τ²) at small aseparation |δ{right arrow over (x)}_(m2,m1)| and τ→0, the reflectivityof i^(th) scatterer in location {right arrow over (x)}_(i)(t+τ) atinstant t+τ for the sensor A_(m2) with center {right arrow over(x)}_(Am2) for (m2≠m1)=1, 2, . . . 6 can be estimated as follows:

$\begin{matrix}{{\Delta\;{n\left\lbrack {{\overset{\rightarrow}{x}}_{{Am}\; 2},{{\overset{\rightarrow}{x}}_{i}\left( {t + \tau} \right)},{t + \tau}} \right\rbrack}} = {{\Delta\; n_{i}} + {\frac{{\partial\Delta}\; n_{i}}{\partial t}\tau} + {\frac{{\partial\Delta}\; n_{i}}{\partial x_{{Am}\; 1}}\delta\; x_{{m\; 2},{m\; 1}}}}} & (19)\end{matrix}$The derivative ∂Δn_(i)/∂t describes the instantaneous rate of change inthe reflectivity of the i^(th) scatterer at instant t for the sensorA_(m1). The rate depends on the rate of change in the scatterer's shape,size, orientation, and/or material. The spatial derivative∂Δn_(i)/∂x_(Am1) in Eq. (19) characterizes the rate of change in thereflectivity of the i^(th) scatterer at instant t with the sensor A_(m1)being moved in x direction. The rate depends on the scatterer's shape,size, orientation, and/or material as well as the scatterer's locationin the volume V at the instant t. The terms with derivatives in Eq. (19)are very small at |δ{right arrow over (x)}_(m2,m1)|/R<<1 and τ→0 in mostatmospheric conditions and they are neglected below.

One can use inequality (5) and equations (11)-(14), (18), (19), andneglect the terms O(τ²) and other insignificant terms to present thepure signal from sensor A_(m2) at the instant t+τ as:

$\begin{matrix}{{S\left( {{\overset{\rightarrow}{x}}_{{Am}\; 2},{t + \tau}} \right)} = {C_{s}{\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\Delta\; n_{i}\Delta\; n_{k}B_{i}B_{k}{\cos\left( {\phi_{ik} - \phi_{ik} + \psi_{ik} - \zeta_{ik}} \right)}}}}}} & (20)\end{matrix}$Here:

$\begin{matrix}{{{\varphi_{ik} = {\varphi_{i} - \varphi_{k}}},{\varphi_{i} = {\frac{2{\pi\gamma\delta}\; x_{{m\; 2},{m\; 1}}}{D}\frac{x_{i}}{\sigma}}}}{{\psi_{ik} = {\psi_{i} - \psi_{k}}},{\psi_{i} = {{4{\pi\gamma}\frac{U_{i}\tau}{D}\frac{x_{i}}{\sigma}} + {4{\pi\gamma}\frac{V_{i}\tau}{D}\frac{y_{i}}{\sigma}} + {4\pi\frac{W_{i}\tau}{\lambda}}}}}{{\zeta_{ik} = {\zeta_{i} - \zeta_{k}}},{\zeta_{i} = {\frac{2{\pi\gamma\delta}\; x_{{m\; 2},{m\; 1}}}{D}\frac{U_{i}\tau}{\sigma}}}}} & (21)\end{matrix}$One can see from equations (15), (16) and (20), (21) that S({right arrowover (x)}_(Am1),t) is independent of {right arrow over (x)}_(Am1). Thesignal S({right arrow over (x)}_(Am2),t+τ) depends only on δ{right arrowover (x)}_(m2,m1) while it is independent of {right arrow over(x)}_(Am1) and {right arrow over (x)}_(Am2), and so are the poweredweighted increments.

The powered weighted increments for a pair of pure signals S({rightarrow over (x)}_(Am1),t) and S({right arrow over (x)}_(Am2),t+τ) can bepresented in a non-dimensional form as follows:

$\begin{matrix}{{{\Phi_{p}\left( {{\delta{\overset{\;\rightarrow}{x}\;}_{{m\; 2},\;{m\; 1}}},\tau,\omega_{x},\omega_{\tau}} \right)}\; = \;{\frac{1}{\left( \sigma_{{Am}\; 1}^{2} \right)^{p/2}}\left\langle \left\{ {\begin{bmatrix}{{S\left( {{\overset{->}{x}}_{{Am}\; 1},t} \right)} -} \\{S\left( {{\overset{->}{x}}_{{Am}\; 1},{t + \tau}} \right)}\end{bmatrix} - {\omega_{x}\begin{bmatrix}{{S\left( {{\overset{->}{x}}_{{Am}\; 1},t} \right)} -} \\{S\left( {{\overset{->}{x}}_{{Am}\; 2},t} \right)}\end{bmatrix}} - {\omega_{\tau}\begin{bmatrix}{{S\left( {{\overset{->}{x}}_{{Am}\; 1},t} \right)} -} \\{S\left( {{\overset{->}{x}}_{{Am}\; 1},{t + \tau}} \right)}\end{bmatrix}}} \right\}^{p} \right\rangle}}{\sigma_{{Am}\; 1}^{2} = \left\langle \left\lbrack {{S\left( {{\overset{->}{x}}_{{Am}\; 1},t} \right)} - \left\langle {S\left( {{\overset{->}{x}}_{{Am}\; 1},t} \right)} \right\rangle} \right\rbrack^{2} \right\rangle}} & (22)\end{matrix}$To compare the instant method with prior art methods, only the secondorder powered weighted increments at p=2 will be considered, and thesubscript p=2 is omitted below. One can substitute the expressions (15)and (20) for pure received signals into Eq. (22), and get the followingequation for the second order powered weighted increments:

$\begin{matrix}{{{\Phi\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\tau,\omega_{x},\omega_{\tau}} \right)} = {\frac{1}{\theta_{0}}\left\langle \;\begin{bmatrix}{\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\Delta\; n_{i}\Delta\; n_{k}B_{i}B_{k}}}} \\\begin{Bmatrix}{\left\lfloor {{\cos\left( \phi_{ik} \right)} - {\cos\left( {\phi_{ik} - \varphi_{ik} + \psi_{ik} - \zeta_{ik}} \right)}} \right\rfloor -} \\{{\omega_{x}\left\lbrack {{\cos\left( \phi_{ik} \right)} - {\cos\left( {\phi_{ik} - \varphi_{ik}} \right)}} \right\rbrack} -} \\{\omega_{\tau}\left\lbrack {{\cos\left( \phi_{ik} \right)} - {\cos\;\left( {\phi_{ik} + \psi_{ik}} \right)}} \right\rbrack}\end{Bmatrix}\end{bmatrix}^{2} \right\rangle}}{\theta_{0} = {\left\langle \left\lbrack {\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\Delta\; n_{i}\Delta\; n_{k}B_{i}B_{k}{\cos\left( \phi_{ik} \right)}}}} \right\rbrack^{2} \right\rangle - \left\langle {\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\Delta\; n_{i}\Delta\; n_{k}B_{i}B_{k}{\cos\left( \phi_{ik} \right)}}}} \right\rangle^{2}}}} & (23)\end{matrix}$

Equations (23), (16), and (21) contain seven random variablescharacterizing each scatterer i=1, 2, . . . , N at instant t: namely,the coordinates x_(i), y_(i), z_(i); the velocity components U_(i),V_(i), W_(i); and the reflectivity Δn_(i). Each one of these variablesdescribes a physically different characteristic of the scatterer,therefore, they can be considered as statistically independent randomvalues for the same and different scatterers. However, the velocitycomponents can be correlated both for the same scatterer and differentscatterers. One can see from definitions (16) and (21) that|φ_(ik)|=O(σ_(r)/λ, σ/D)>>1 are very large, |φ_(ik)|=O(1) are of theunity order while |ψ_(ik)|, |ζ_(ik)|=O(τ)<<1 and|ζ_(ik)|/|ψ_(ik)|=O(U)_(i)τ/σ)<<1 are very small at τ→0. Using theseconsiderations and applying lengthy but straightforward mathematicalmanipulations, one can reduce equations (23) to the following form:

$\begin{matrix}{{{\Phi\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\tau,\omega_{x},\omega_{\tau}} \right)} = {\frac{1}{\theta_{0}}{\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\left\langle {\Delta\; n_{i}} \right\rangle\left\langle {\Delta\; n_{k}} \right\rangle\left\langle {B_{i}^{2}B_{k}^{2}\begin{Bmatrix}\begin{matrix}{{2\left( {1 - \omega_{x}} \right)^{2}\left\lfloor {1 - {\cos\left( \varphi_{ik} \right)}} \right\rfloor} - {2\left( {1 - \omega_{x}} \right)\left( {1 - \omega_{\tau}} \right)}} \\{{\psi_{ik}{\sin\left( \varphi_{ik} \right)}} + {\left( {1 - \omega_{x} - \omega_{\tau}} \right)\left( {\psi_{ik} - \zeta_{ik}} \right)^{2}{\cos\left( \varphi_{ik} \right)}} +}\end{matrix} \\{{\omega_{x}\omega_{\tau}\psi_{ik}^{2}{\cos\left( \varphi_{ik} \right)}} - {\omega_{x}\left( {\psi_{ik} - \zeta_{ik}} \right)}^{2} - {{\omega_{\tau}\left( {1 - \omega_{x} - \omega_{\tau}} \right)}\psi_{ik}^{2}}}\end{Bmatrix}} \right\rangle}}}}}{\theta_{0} = {\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\left\langle {\Delta\; n_{i}^{2}} \right\rangle\left\langle {\Delta\; n_{k}^{2}} \right\rangle\left\langle B_{i}^{2} \right\rangle\left\langle B_{k}^{2} \right\rangle}}}}} & (24)\end{matrix}$If one takes into account that |ψ_(ik)|, |ζ_(ik)| ∝ τ, at very smallτ→0, the powered weighted increments (24) can be presented as follows:

$\begin{matrix}{\;{{{\Phi\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\tau,\omega_{x},\omega_{\tau}} \right)} = {{d_{0}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)} + {\eta\;{d_{1}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)}} + {\eta^{2}{d_{2}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)}} + {O\left( \eta^{3} \right)}}},\mspace{14mu}{\eta\; = {{\tau/\delta}\; t}}}} & (25)\end{matrix}$The values Φ, d₀, d₁, d₂, and η in this equation are non-dimensional,and the coefficients d₀, d₁, and d₂ for the pure signals are as follows:

$\begin{matrix}{{{d_{0}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)} = {\frac{2\left( {1 - \omega_{x}} \right)^{2}}{\theta_{0}}{\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\left\langle {\Delta\; n_{i}^{2}} \right\rangle\left\langle {\Delta\; n_{k}^{2}} \right\rangle\left\langle {B_{i}^{2}{B_{k}^{2}\left\lbrack {1 - {\cos\left( \varphi_{ik} \right)}} \right\rbrack}} \right\rangle}}}}}{{d_{1}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)} = {{- \frac{2\left( {1 - \omega_{x}} \right)\left( {1 - \omega_{\tau}} \right)}{\theta_{0}}}{\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\left\langle {\Delta\; n_{i}^{2}} \right\rangle\left\langle {\Delta\; n_{k}^{2}} \right\rangle\left\langle {B_{i}^{2}B_{k}^{2}\psi_{ik}{\sin\left( \varphi_{ik} \right)}} \right\rangle}}}}}{d_{2}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)} = {\frac{1}{\theta_{0}}{\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\left\langle {\Delta\; n_{i}^{2}} \right\rangle\left\langle {\Delta\; n_{k}^{2}} \right\rangle\left\langle {B_{i}^{2}B_{k}^{2}\begin{Bmatrix}{{\left. {1 - \omega_{x} - \omega_{\tau}} \right)\left( {\psi_{ik} - \zeta_{ik}} \right)^{2}{\cos\left( \varphi_{ik} \right)}} +} \\{{\omega_{x}\omega_{\tau}\psi_{ik}^{2}{\cos\left( \varphi_{ik} \right)}} -} \\{{\omega_{x}\left( {\psi_{ik} - \zeta_{ik}} \right)}^{2} - {{\omega_{\tau}\left( {1 - \omega_{x} - \omega_{\tau}} \right)}\psi_{ik}^{2}}}\end{Bmatrix}} \right\rangle}}}}} & (26)\end{matrix}$It is important to note that equations (24)-(26) are very generic. Theyare derived for a volume-scattering model using only assumption 1. It isalso important that each coefficient in the decomposition (25), (26)describes physically different characteristics of the monitored object.One can see from (16), (19), and (21) that for a particular case ofatmospheric scatterers, d₀ depends on their size, shape, content andspatial distribution inside the predetermined volume V although it isfully independent of the scatterer's motion. It is shown below that d₁depends on the mean speed of the atmosphere in the volume V although itis independent of turbulence. Atmospheric turbulence affects onlycoefficient d₂.

Identification of the Monitored Object

Coefficient d₀ in Eq. (26) can be further reduced to the following form:

$\begin{matrix}{{d_{0}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)} = {2\left( {1 - \omega_{x}} \right)^{2}{\left\{ {1 - {\frac{1}{\theta_{0}}{\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\left\langle {\Delta\; n_{i}^{2}} \right\rangle\left\langle {\Delta\; n_{k}^{2}} \right\rangle \times \left\lfloor \begin{matrix}{{\left\langle {B_{i}^{2}{\cos\left( \varphi_{i} \right)}} \right\rangle\left\langle {B_{k}^{2}{\cos\left( \varphi_{k} \right)}} \right\rangle} +} \\{\left\langle {B_{i}^{2}{\sin\left( \varphi_{i} \right)}} \right\rangle\left\langle {B_{k}^{2}{\sin\left( \varphi_{k} \right)}} \right\rangle}\end{matrix} \right\rfloor}}}}} \right\}.}}} & (27)\end{matrix}$One can see from definitions (16) and (21) that B_(i) and φ_(i) containan infinite number of unknown variables: the instantaneous reflectivityΔn_(i)(t) and coordinates x_(i)(t), y_(i)(t) and z_(i)(t) of eachscatterer i=1, 2, . . . , N. In a general case, Eq. (27) also containsthe instantaneous derivatives from Eq. (19) that are neglected in thegiven example of the profiling radar based system. The reflectivityΔn_(i)(t) and its derivatives ∂n_(i)(t)/∂t, ∂n_(i)(t)/∂x_(Am1) depend onthe scatterer's size, shape, content, orientation as well as the rate ofchanges in the above parameters while the coordinates x_(i)(t), y_(i)(t)and z_(i)(t) depend on the spatial distribution of scatterers inside theilluminated volume. Therefore, coefficient d₀ provides indicators foridentifying the monitored object in the present system.

However, Eq. (27) cannot be used directly in practical measurementsbecause of an infinite number of unknown variables. To get anoperational equation, one should make specific assumptions, choose amore specific model for a monitored object, or both. As an example, onecan consider a single thermal plume in the center of the illuminatedvolume with the Gaussian spatial distribution of scatterers inside theplume:

$\begin{matrix}{{{P_{{m\; 2},{m\; 1}}\left( x_{i} \right)} = {\frac{1}{\sqrt{2\pi}\sigma_{{m\; 2},{m\; 1}}}{\exp\left( {- \frac{x_{i}^{2}}{2\sigma_{{m\; 2},{m\; 1}}^{2}}} \right)}}},{i = {1,2,\mspace{11mu}\ldots}}\mspace{11mu},N} & (28)\end{matrix}$Here P_(m1,m2)(x_(i)) is the probability density distribution of thei^(th) scatterer location along an arbitrary horizontal baseline δ{rightarrow over (x)}_(m2,m1) considered in the present system, andσ_(m2,m1)<<σ is the plume's width in the direction δ{right arrow over(x)}_(m2,m1); the latter is an unknown parameter of the object to bedetermined in this example. One can combine equations (16), (21), (27)and (28) and derive the following equation:

$\begin{matrix}{{d_{0}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)} = {2\left( {1 - \omega_{x}} \right)^{2}\left\lfloor {1 - {\exp\left( {{- \frac{4\pi^{2}\gamma^{2}\delta\; x_{{m\; 2},{m\; 1}}^{2}}{D^{2}}}\frac{\sigma_{{m\; 2},{m\; 1}}^{2}}{\sigma^{2}}} \right)}} \right\rfloor}} & (29)\end{matrix}$It follows from Eqs. (29) and (13) that:

$\begin{matrix}{\sigma_{{m\; 2},{m\; 1}} = {\sqrt{{- \ln}\left\{ {1 - {{d_{0}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)}/\left\lbrack {2\left( {1 - \omega_{x}} \right)^{2}} \right\rbrack}} \right\}}\frac{\lambda}{\delta\; x_{{m\; 2},{m\; 1}}}\frac{R}{2\pi}}} & (30)\end{matrix}$Eq. (30) is an example of an operational equation for the given example.It analytically relates the selected characteristic of the monitoredobject, the plume width σ_(m2,m1), to the coefficient d₀ in thedecomposition (25) of measured powered weighted increments for a pair ofsensors A_(m1) and A_(m2), (m2≠m1)=1, 2, . . . 6. The width σ_(m2,m1) ofthe thermal plume can be estimated along all available baselines(A₁,A₂), (A₂,A₃), (A₁,A₃) as well as (A₁,A₅), (A₂,A₆), (A₃,A₄) becaused₀ does not depend on α in this case. The values of σ_(m2,m1)comprehensively characterize the size and shape of the plume although itis much smaller than the size σ of the illuminated volume. This simpleexample illustrates the major advantages of the present system andmethod.

First, Eq. (30) had been derived with the method of the presentinvention for a specified object given by Eq. (28) using the only onenatural and generic assumption 1. The existing methods would require alarger number of more restrictive assumptions for the derivation.Second, one can get multiple estimates for each measured value σ_(m2,m1)by varying the weights ω_(x) and ω_(τ). For example, the weights ω_(x)=0and −1 employ only the first and two first terms in the powered weightedincrements (22); the value of d₀ is 4 times larger in the latter case.In addition, one can use both values of ω_(x)=0 and −1 at differentω_(τ), for example, ω_(τ)=0 and 1. These simple combinations aloneprovide 4 estimates for each σ_(m2,m1). One can further double thenumber of estimates using parallel baselines, for example, (A₁,A₂) and(A₅,A₆). The importance of multiple estimates and their practicalapplications are discussed below in the detailed description of step106.

Measuring the Mean Speed of the Monitored Object

Coefficient d₁ in Eq. (26) can be further reduced to the following form:

$\begin{matrix}{{d_{1}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)} = {{- \frac{4\left( {1 - \omega_{x}} \right)\left( {1 - \omega_{\tau}} \right)}{\theta_{0}}}{\sum\limits_{i = 1}^{N}{\sum\limits_{k = 1}^{N}{\left\langle {\Delta\; n_{i}^{2}} \right\rangle\left\langle {\Delta\; n_{k}^{2}} \right\rangle \times \left\lfloor \begin{matrix}{{\left\langle {B_{i}^{2}\psi_{i}{\sin\left( \varphi_{i} \right)}} \right\rangle\left\langle {B_{k}^{2}\cos\left( \varphi_{k} \right)} \right\rangle} -} \\{\left\langle {B_{i}^{2}\psi_{i}{\cos\left( \varphi_{i} \right)}} \right\rangle\left\langle {B_{k}^{2}{\sin\left( \varphi_{k} \right)}} \right\rangle}\end{matrix} \right\rfloor}}}}} & (31)\end{matrix}$Similarly to Eq. (27), Eq. (31) contains an infinite number of unknownvariables. In addition to the instantaneous reflectivity Δn_(i)(t) andcoordinates x_(i)(t), y_(i)(t) and z_(i)(t), a generic Eq. (31) for d₁contains the instantaneous velocity components U_(i)(t), V_(i)(t) andW_(i)(t) for each scatterer i=1, 2, . . . , N. To turn Eq. (31) into anoperational equation, one should make specific assumptions, choose amore specific model for a monitored object, or both. For the givenexample of the atmospheric profiling radar based system, two rathergeneric and natural assumptions are sufficient for deriving theoperational equation.

Assumption 2: The spatial distribution of scatterers inside theilluminated volume V is statistically uniform in any horizontaldirection; that is:P _(m2,m1)(x _(i))=const, i=1, 2, . . . , N   (32)

Assumption 3: All scatterers in the illuminated volume move with thesame mean horizontal velocity components; that is:<U _(i) >=<U>, <V _(i) >=<V>, i=1, 2, . . . , N   (33)Note that assumption 2 replaces the model (28) that was used in theprevious subsection. One can combine equations (16), (17), (21), (27),and (31)-(33) to derive the following expressions:

$\begin{matrix}{{{d_{0}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)} = {2\left( {1 - \omega_{x}} \right)^{2}\left( {1 - e_{{m\; 2},{m\; 1}}} \right)}}{{d_{1}\left( {{\delta\;{\overset{\rightarrow}{x}}_{{m\; 2},{m\; 1},}\omega_{x}},\omega_{\tau}} \right)} = {{- 8}\left( {1 - \omega_{x}} \right)\left( {1 - \omega_{\tau}} \right)\begin{matrix}{\frac{\left\langle U_{{m\; 2},{m\; 1}} \right\rangle\delta\; t}{g}\frac{\delta\; x_{{m\; 2},{m\; 1}}}{g}} \\e_{{m\; 2},{m\; 1}}\end{matrix}}}{{e_{{m\; 2},{m\; 1}} = {\exp\left( {- \frac{\delta\; x_{{m\; 2},{m\; 1}}^{2}}{g^{2}}} \right)}},{g = {\frac{\alpha}{2{\pi\gamma}}D}}}} & (34)\end{matrix}$One can see that coefficient d₁ depends only on the mean velocity<U_(m2,m1)> although it is independent of turbulence. One should recallthat equations (19), (21), (24, and (26) are derived for the adaptivecoordinate system with the x axis along the baseline δ{right arrow over(x)}_(m2,m1). Notations σ_(m2,m1), <U_(m2,m1)> and similar notationsbelow emphasize that the measured values are those along δ{right arrowover (x)}_(m2,m1), for example, <U_(m2,m1)> is the projection of themean velocity of atmospheric scatterers on the direction δ{right arrowover (x)}_(m2,m1). It follows from equations (34) that:

$\begin{matrix}{{\left\langle U_{{m\; 2},{m\; 1}} \right\rangle = {{- \frac{d_{1}\left( {{\delta\;{\overset{->}{x}}_{{m\; 2},{m\; 1}}},{\omega_{x}\omega_{\tau}}} \right)}{8\left( {1 - \omega_{x}} \right)\left( {1 - \omega_{\tau}} \right)\mu_{0}{\ln\left( \mu_{0} \right)}}}\frac{\delta\; x_{{m\; 2},{m\; 1}}}{\delta\; t}}}\mu_{0} = {1 - {{d_{0}\left( {{\delta\;{\overset{->}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)}/\left\lbrack {2\left( {1 - \omega_{x}} \right)^{2}} \right\rbrack}}} & (35)\end{matrix}$Eq. (35) is another example of an operational equation for the givenexample of implementing the present invention. It analytically relatesthe selected characteristic of the monitored object, a projection of themean velocity of atmospheric scatterers on the direction δ{right arrowover (x)}_(m2,m1), to the coefficients d₀ and d₁ in the decomposition(25) of measured powered weighted increments for a pair of sensorsA_(m1) and A_(m2), (m2≠m1)=1, 2, . . . 6. One can measure projections ofthe mean horizontal velocity on different baselines δ{right arrow over(x)}_(m2,m1), (m2≠m1)=1, 2, . . . 6 with the present system. Any twonon-parallel baselines are sufficient for estimating the mean horizontalvelocities <U> and <V> in any specified coordinate system, for example,a geophysical system with the x axis extending towards the east and they axis extending towards the north. With any two non-parallel baselines,one can therefore estimate the mean horizontal speed <V_(h)> which isinvariant of the coordinate system:V _(h) =√{square root over (<U> ² +<V> ²)}  (36)One can consider the baselines (A₁,A₂), (A₂,A₃), (A₁,A₃) and (A₄,A₅),(A₅,A₆), (A₄,A₆), However, one cannot apply Eq. (35) to mixed pairs like(A₁,A₅), (A₂,A₆), or (A₃,A₄). To simplify the equations, the same valueof α within a pair was considered above; equations for different α canbe easily derived in a similar way.

The example further illustrates the major advantages of the presentsystem and method. First, multiple estimates for the projection<U_(m2,m1)> on each baseline, that is for a single pair of sensors canbe obtained by varying the weights. The most efficient combinations forthe present system are ω_(x)=ω_(τ)=0 and ω_(x)=ω_(τ)=−1. Second, Eq.(35) had been derived with the method of the present invention usingnatural and generic assumptions 1, 2 and 3. The existing methods wouldrequire a larger number of more restrictive assumptions for thederivation. For example, the most advanced correlation function-basedtechnique for spaced antenna radars, the Holloway—Doviak method requireseight significantly more restrictive assumptions for deriving the onlyone operational equation for a single pair of sensors.

Equation (35) provides a projection of the mean velocity of atmosphericscatterers on the direction δ{right arrow over (x)}_(m2,m1). In theexample of the system configuration in FIGS. 2 b and 4, all physicalsensors A₁, A₂, A₃ and, hence virtual sensors A₄, A₅, A₆ are separatedonly in the horizontal directions. Therefore, one cannot measure themean vertical velocity of atmospheric scatterers <W> with the presentedsystem configuration using the present method. There are two ways formeasuring <W> with the exemplary embodiment shown in FIGS. 2 and 3.First, one can apply the standard Doppler spectra to the complexreceived signals given by Eq. (4). Second, one can adjust the systemconfiguration for applying the method of the present invention. Forexample, one can position at least one physical sensor A_(m), m=1, 2, or3 in FIG. 2B at a different height with respect to other sensors.Operational equations for measuring <W> using a pair of sensorsseparated in the vertical direction can be easily derived with themethod of the present invention in a manner similar to deriving Eq.(35).

Measuring the Rates of Changes in the Monitored Object

The rates of changes depend on the monitored object and specificequations for measuring the rates also depend on the type andconfiguration of monitoring equipment. For the given example of theatmospheric profiling radar based system, the monitored object is theatmosphere in the predetermined volume, and the rates mainly depend onatmospheric turbulence that changes the relative location of scattererswith respect to each other. One can see from equations (29) and (34)that turbulence does not affect coefficients d₀ and d₁ in decomposition(25). On the contrary, coefficient d₂ is heavily dependent onturbulence. To derive an operational equation for d₂, it is sufficientto apply assumptions 1, 2, 3, and one more rather generic assumption.

Assumption 4: The statistical characteristics of turbulent motion arethe same for all scatterers in the illuminated volume; that is:<u_(i) ²>=<u²>, <v_(i) ²>=<v²>, <w_(i) ²>=<w²><u_(i)v_(i)>=<uv>, <u_(i)w_(i)>=<uw>, <v_(i)w_(i)>=<vw>, i=1, 2, . . . ,N   (37)One can combine equations (16), (17), (21), (26), (32), (33), (36), and(37) to derive the following expression:

$\begin{matrix}{{\begin{matrix}{{d_{2}\left( {{\delta\;{\overset{->}{x}}_{{m\; 2},{m\; 1}}},\omega_{x},\omega_{\tau}} \right)} = {8\left\lbrack {{\left( {{4\pi^{2}\frac{\left\langle w^{2} \right\rangle\delta\; t^{2}}{\lambda^{2}}} + \frac{\left\langle V_{h} \right\rangle^{2}\delta\; t^{2}}{g^{2}}} \right)\mu_{w}} -} \right.}} \\{\frac{\left( {{2\left\langle U_{{m\; 2},{m\; 1}} \right\rangle^{2}} + \left\langle u_{{m\; 2},{m\; 1}}^{2} \right\rangle} \right)\delta\; t^{2}}{g^{2}}\frac{\delta\; x_{{m\; 2},{m\; 1}}^{2}}{g^{2}}} \\{{e_{{m\; 2},{m\; 1}}\mu_{u}} - {2{\pi\alpha}\frac{\left\langle {u_{{m\; 2},{m\; 1}}w} \right\rangle\delta\; t^{2}}{\sigma\lambda}}} \\\left. {\frac{\delta\mspace{11mu} x_{\;{{m\; 2},\;{m\; 1}}}}{\; g}\mu_{\;{uw}}} \right\rfloor\end{matrix}{\mu_{w} = {\mu_{u} - {\left( {1 - \omega_{\tau}} \right)\left( {\omega_{x} + \omega_{\tau}} \right)}}},{\mu_{u} = {\left( {1 - \omega_{x}} \right)\left( {1 - \omega_{\tau}} \right)e_{{m\; 2},{m\; 1}}}}}{\mu_{uw} = {{\left( {1 - \omega_{\tau}} \right)e_{{m\; 2},{m\; 1}}} - {\left( {1 + e_{{m\; 2},{m\; 1}}} \right)\omega_{x}}}}} & (38)\end{matrix}$The mean horizontal velocities <V_(h)> and <U_(m2,m1)> are estimatedindependently using coefficients d₀ and d₁ hence Eq. (38) contains onlythree unknown values: <w²>, <u_(m2,m1) ²>, and <u_(m2,m1)w>. One can getmultiple estimates for all characteristics of atmospheric turbulence byapplying Eq. (38) to different pairs of sensors A_(m1), A_(m2) andvarying the weights ω_(x) and ω_(τ) for each selected pair. Numerouscombinations of ω_(x) and ω_(τ) provide numerous equations forestimating the unknowns. However, a blind approach is inefficient andits accuracy could be extremely poor. One can see that theright-hand-side terms in Eq. (38) dramatically differ in magnitude. Thefirst two terms with a multiplier μ_(w) are much larger than the thirdone with a multiplier μ_(u), and the third term is much larger than thefourth with a multiplier μ_(uw). As demonstrated below, the poweredweighted increments enable one to overcome this problem by selectingspecific weights ω_(x) and ω_(τ) for each selected pair of sensorsA_(m1), A_(m2) and to ensure the most efficient and accurate measurementof each characteristic of turbulence.

The variance of the vertical turbulent velocity <w²> can be accuratelyestimated using each sensor separately; that is at δ{right arrow over(x)}_(m2,m1)=0, e_(m2,m1)=1. Without losing generality, one canprescribe ω_(x)=0 because d₂ is independent of ω_(x) in this case. Onecan then estimate <w²> with the following equation:

$\begin{matrix}{\left\langle w^{2} \right\rangle = {{\frac{d_{2}\left( {0,0,\omega_{\tau}} \right)}{32{\pi^{2}\left( {1 - \omega_{\tau}} \right)}^{2}}\frac{\lambda^{2}}{\delta\; t^{2}}} - {\frac{\lambda^{2}}{4\pi^{2}g^{2}}\left\langle V_{h} \right\rangle^{2}}}} & (39)\end{matrix}$The most fruitful weights for estimating <w²> with Eq. (39) areω_(τ)=−1, 0 and ½.

To get reliable and accurate estimates for <u_(m2,m1) ²>, one should getrid of the first two terms in the right-hand-side of Eq. (38). It can beeasily achieved by selecting the weights that satisfy the equationμ_(w)=0 and inequality μ_(u)≠0, that is:(1−ω_(r))(e _(m2,m1)−ω_(τ)−ω_(x)−ω_(x) e _(m2,m1))=0, ω_(x)≠1, ω_(τ)≠1  (40)Solutions of this equation are as follows:

$\begin{matrix}{{\omega_{x} = \frac{e_{{m\; 2},{m\; 1}} - \omega_{\tau}}{1 + e_{{m\; 2},{m\; 1}}}},{\omega_{\tau} \neq 1},{- 1}} & (41)\end{matrix}$One can get multiple reliable and accurate estimates of <u_(m2,m1) ²>with any pair of weights ω_(x) and ω_(τ) satisfying equation (41)because the third term in Eq. (38) is much larger than the fourth one.An elegant estimation can be made at ω_(τ=e) _(m2,m1) which givesω_(x)=0. In this case:

$\begin{matrix}{\left\langle u_{{m\; 2},{m\; 1}}^{2} \right\rangle = {{{- \frac{d_{2}\left( {{\delta\;{\overset{->}{x}}_{{m\; 2},{m\; 1}}},0,e_{{m\; 2},{m\; 1}}} \right)}{8{e_{{m\; 2},{m\; 1}}\left( {1 - e_{{m\; 2},{m\; 1}}} \right)}}}\frac{g^{2}}{\delta\; x_{{m\; 2},{m\; 1}}^{2}}\frac{g^{2}}{\delta\; t^{2}}} - {2\left\langle U_{{m\; 2},{m\; 1}} \right\rangle^{2}}}} & (42)\end{matrix}$One can apply Eq. (42) to any three non-parallel baselines, for example(A₁,A₂), (A₂,A₃), (A₁,A₃) and (A₄,A₅), (A₅,A₆), (A₄,A₆), and getestimates for the variances of the horizontal turbulent velocities <u²>,<v²> and the horizontal momentum flux <uv> with the standardtrigonometric relations in any specified Cartesian coordinate system.

The most difficult task is to estimate a projection of the vertical fluxbecause the term with <u_(m2,m1)w> in the right-hand-side of Eq. (38) ismuch smaller than the others. The only way of getting reliable andaccurate estimates is to get rid of all terms in the right-hand-side ofEq. (38) except for the last one. To do so, one should select suchweights ω_(x) and ω_(τ) that would satisfy three conditions:μ_(w)=0, μ_(u)=0, μ_(uw)≠0   (43)The conditions can be satisfied by choosing the following weights:ω_(τ)=1, ω_(x)≠0   (44)One can get numerous estimates of the flux <u_(m2,m1)w> by varying theweight ω_(x). The most elegant solution is at ω_(x)=1 and it is asfollows:

$\begin{matrix}{\left\langle {u_{{m\; 2},{m\; 1}}w} \right\rangle = {\frac{d_{2}\left( {{\delta\;{\overset{->}{x}}_{{m\; 2},{m\; 1}}},1,1} \right)}{16{{\pi\alpha}\left( {1 + e_{{m\; 2},{m\; 1}}} \right)}}\frac{g}{\delta\; x_{{m\; 2},{m\; 1}}}\frac{\sigma\lambda}{\delta\; t^{2}}}} & (45)\end{matrix}$Note that d₀=d₁=0 at ω_(x)=1 which further increases the accuracy ofestimating <u_(m2,m1)w> with Eq. (45). One can apply Eq. (45) to any twonon-parallel baselines and get multiple reliable and accurate estimatesfor the variances of the vertical momentum fluxes <uw> and <vw> with thestandard trigonometric relations in any specified Cartesian coordinatesystem.

Eqs. (39), (42), and (45) are other examples of operational equationsfor the given example of implementing the present invention. Theequations analytically relate the selected characteristics of themonitored object, namely, the characteristics of atmospheric turbulence<w²>, <u_(m2,m1) ²>, and <u_(m2,m1)w> to the coefficients d₀, d₁, and d₂in the decomposition (25) of measured powered weighted increments for apair of sensors A_(m1) and A_(m2), (m2≠m1)=1, 2, . . . 6. Theseequations clearly demonstrate two remarkable features that distinguishthe present, powered weighted increments-based method from all prior artmethods: one can get numerous operational equations for each pair ofsensors A_(m1), A_(m2) by varying the weights ω_(x) and ω_(τ), and onecan select specific weights for each selected pair of sensors for themost efficient and accurate measurement of each predeterminedcharacteristic of the monitored object. These two features also enablethe measurement of characteristics of the monitored object that cannotbe measured with prior art methods. For example, none of the prior artmethods can provide the fluxes <uw> and <vw>.

Signals with Noise

Operational equations (30), (35), (39), (42), and (45) were derived forpure signals from atmospheric scatterers in the illuminated volume. Theright-hand-side of these equations contain characteristics of theatmosphere that are intended to be estimated with the present system andmethod for measuring characteristics of continuous medium and/orlocalized targets. The equations contain coefficients d₀, d₁, and d₂ ofthe powered weighted increments of the pure signal power with no noiseor other contaminants; the latter are always present in the actual andcombined signals. For the equations to be applicable to practicalmeasurements, one should relate the powered weighted increments for puresignals (22) to those for signals with noise (9).

It is well known that any differential data processing method is notaffected by clutter, low-frequency radio interference, or othercontaminants with sufficiently large temporal scale T_(cor). One canconclude from equations (2) and (3) that the powered weighted incrementsbelong to the class of differential processing tools and, therefore areunaffected by contaminants with large T_(cor)>>δt. However, differentialtools such as structure functions and powered weighted increments arehighly sensitive to white noise with very small or zero temporal scale.For this reason, only white noise n({right arrow over (x)}_(Am),t) inEq. (7) is considered below, and the standard assumption about suchnoise is the following.

Assumption 5: The pure received signal S({right arrow over (x)}_(Am),t)and the noise n({right arrow over (x)}_(Am),t) are uncorrelated.

This assumption and a definition of white noise can be formalized asfollows:<S({right arrow over (x)} _(Am1) ,t)n({right arrow over (x)} _(Am1),t+τ)>=<S({right arrow over (x)} _(Am1) ,t)n({right arrow over (x)}_(Am2) ,t+τ)>=0 at any τ<n({right arrow over (x)} _(Am1) ,t)n({right arrow over (x)} _(Am1),t+τ)>=<n({right arrow over (x)} _(Am1) ,t)n({right arrow over (x)}_(Am2) ,t+τ)>=0 at τ≠0   (46)One can combine equations (7), (22), and (46) with Eq. (9) at p=2 toderive the following expression where δ(τ) is the Kroneker function:

$\begin{matrix}\begin{matrix}{\begin{matrix}{\sigma_{\;{{m\; 2},\;{m\; 1}}}^{\; 2}\Phi} \\\left( {{\delta\;{\overset{->}{x}}_{{m\; 2},{m\; 1}}},\tau,\omega_{x},\omega_{\tau}} \right)\end{matrix} = {{\overset{\sim}{\Phi}\left( {{\delta\;{\overset{->}{x}}_{{m\; 2},{m\; 1}}},\tau,\omega_{x},\omega_{\tau}} \right)} -}} \\{\left( {1 + \omega_{x}^{2} + {2\omega_{\tau}^{2}} - {2\omega_{x}} - {2\omega_{\tau}} + {2\omega_{x}\omega_{\tau}}} \right)} \\{\left\langle {n^{2}\left( {\overset{->}{x}}_{{Am}\; 1} \right)} \right\rangle - {\left( {1 + \omega_{x}^{2}} \right)\left\langle {n^{2}\left( {\overset{->}{x}}_{{Am}\; 2} \right)} \right\rangle} +} \\{2{\delta(\tau)}\left( {\omega_{x}^{2} + \omega_{x} + \omega_{\tau} + {\omega_{x}\omega_{\tau}}} \right)} \\{\left\langle {{n\left( {\overset{->}{x}}_{{Am}\; 1} \right)}{n\left( {\overset{->}{x}}_{{Am}\; 2} \right)}} \right\rangle}\end{matrix} & (47)\end{matrix}$Eq. (47) relates the powered weighted increments of pure signals tothose of signals with noise; the latter can be calculated directly forreceived signals with Eq. (9). However, the equation contains three newunknown variables: the second order moments of noise <n²({right arrowover (x)}_(Am1))>, <n²({right arrow over (x)}_(Am1))> and thecorrelation <n({right arrow over (x)}_(Am1))n({right arrow over(x)}_(Am2))>. One can see that all terms with noise in Eq. (47) areindependent of τ hence they contribute only to the coefficients {tildeover (d)}₀(δ{right arrow over (x)}_(m2,m1),ω_(x),ω_(τ)) in Eq. (10)while they affect neither coefficients {tilde over (d)}₁(δ{right arrowover (x)}_(m2,m1),ω_(x),ω_(τ)) nor {tilde over (d)}₂(δ{right arrow over(x)}_(m2,m1),ω_(x),ω_(τ)). It follows from Eq. (47) that:

$\begin{matrix}\begin{matrix}{{{\overset{\sim}{d}}_{0}\left( {{\delta\;{\overset{->}{x}}_{{m\; 2},{m\; 1}}},{\omega_{x,}\omega_{\tau}}} \right)} = {{\sigma_{{Am}\; 1}^{2}{d_{0}\left( {{\delta\;{\overset{->}{x}}_{{m\; 2},{m\; 1}}},\tau,\omega_{x},\omega_{\tau}} \right)}} +}} \\{\left( {1 + \omega_{x}^{2} + {2\omega_{\tau}^{2}} - {2\omega_{x}} - {2\omega_{\tau}} + {2\omega_{x}\omega_{\tau}}} \right)} \\{\left\langle {n^{2}\left( {\overset{->}{x}}_{{Am}\; 1} \right)} \right\rangle + {\left( {1 + \omega_{x}^{2}} \right)\left\langle {n^{2}\left( {\overset{->}{x}}_{{Am}\; 2} \right)} \right\rangle} -} \\{2{\delta(\tau)}\left( {\omega_{x}^{2} + \omega_{x} + \omega_{\tau} + {\omega_{x}\omega_{\tau}}} \right)} \\{\left\langle {{n\left( {\overset{->}{x}}_{{Am}\; 1} \right)}{n\left( {\overset{->}{x}}_{{Am}\; 2} \right)}} \right\rangle}\end{matrix} & (48)\end{matrix}$It follows from a generic Eq. (26) that d₀≡0 at ω_(x)=1 which gives:{tilde over (d)} ₀(δ{right arrow over (x)} _(m2,m1),1,ω_(τ))=2ω_(τ) ² <n²({right arrow over (x)} _(Am1))>+2<n ²({right arrow over (x)}_(Am2))>−4δ(τ)(1+ω_(τ)<) n({right arrow over (x)} _(Am1))n({right arrowover (x)} _(Am2))>  (49)One can get numerous independent linear equations for <n²({right arrowover (x)}_(Am1))>, <n²({right arrow over (x)}_(Am1))> and <n({rightarrow over (x)}_(Am1))n({right arrow over (x)}_(Am2))> by varying theweight ω_(τ) in Eq. (49). For example, the equations at ω_(τ)=1, 0, and−1 are as follows:<n ²({right arrow over (x)} _(Am1))>+<n ²({right arrow over (x)}_(Am2))>−4<n({right arrow over (x)} _(Am1))n({right arrow over (x)}_(Am2))>={tilde over (d)} ₀(δ{right arrow over (x)} _(m2,m1),1,1)/2, τ=0<n ²({right arrow over (x)} _(Am1))>+<n ²({right arrow over (x)}_(Am2))>={tilde over (d)} ₀(δ{right arrow over (x)} _(m2,m1),1,1)/2, τ≠0<n ²({right arrow over (x)} _(Am2))>−2<n({right arrow over (x)}_(Am1))n({right arrow over (x)} _(Am2))>={tilde over (d)} ₀(δ{rightarrow over (x)} _(m2,m1),1,0)/2, τ=0<n ²({right arrow over (x)} _(Am2))>={tilde over (d)} ₀(δ{right arrowover (x)} _(m2,m1),1,0)/2, τ≠0<n ²({right arrow over (x)} _(Am1))>+<n ²({right arrow over (x)}_(Am2))>={tilde over (d)} ₀(δ{right arrow over (x)} _(m2,m1),1,−1)/2,τ=0, τ≠0  (50)Therefore, one can get five independent linear equations (50) from Eq.(49) using just three weights ω_(τ)=1, 0, and −1. These equations aremore than sufficient for reliably estimating three unknown values<n²({right arrow over (x)}_(Am1))>, <n²({right arrow over (x)}_(Am1))>and <n({right arrow over (x)}_(Am1))n({right arrow over (x)}_(Am2))> fora pair of sensors A_(m1) and A_(m2).

Eqs. (50) are also examples of operational equations for the givenexample of implementing the present invention. The equationsanalytically relate the characteristic of noise <n²({right arrow over(x)}_(Am1))>, <n²({right arrow over (x)}_(Am1))> and <n({right arrowover (x)}_(Am1))n({right arrow over (x)}_(Am2))> for a pair of sensorsA_(m1) and A_(m2) to the coefficients {tilde over (d)}₀ in thedecomposition of measured powered weighted increments for actual andcombined signals with noise. Again, no other method enables estimatingthe correlation of noise <n({right arrow over (x)}_(Am1))n({right arrowover (x)}_(Am2))> for a pair of sensors.

Analysis of Estimated Characteristics

Multiple estimates for each selected characteristic of the monitoredobject can be analyzed at step 106. Such an analysis may be performed toprovide the best estimate for each characteristic, to obtain a measureof the accuracy for the best estimate, and, if required, to provide ameasure for the reliability of the best estimates. The analysis caninclude (although it is not limited by) the following operations.

(a) A statistical analysis of all obtained estimates for the selectedcharacteristics of the monitored object at each analyzed average timeinterval T_(av) and range R. This important operation utilizes anoutstanding ability of the powered weighted increments to providemultiple equations for each pair of sensors and hence multiple estimatesfor each selected characteristic of the monitored object. For example,one can obtain 12 estimates for the mean horizontal velocities <U> and<V> in the geophysical coordinate system with Eq. (35) using the sensorpairs (A₁,A₂), (A₂,A₃), (A₁,A₃) and (A₄,A₅), (A₅,A₆), (A₄,A₆) and twocombinations of weights ω_(x)=ω_(τ)=0 and ω_(x)=ω_(τ)=−1. Each othercombination of weights gives 6 more estimates. Theoretically all of theestimates should be identical although it is never the case in practicalmeasurements due to a local violation of some assumptions, noise,outliers in the received signals, and many other reasons. Multipleestimates for each interval T_(av) and range R are random samples of thecharacteristic that form a statistical ensemble. Statisticalcharacteristics of the ensemble such as the mean (or median) value andthe standard deviation (or specified percentile points) provide,respectively the best estimate for the characteristic and itsexperimental measurement error. The latter is a metric for measurementaccuracy. An ensemble of multiple estimates for each characteristic alsoallows significant improvement in the temporal resolution of the systembecause averaging over the ensemble is equivalent to additional temporalaveraging. The ability to provide the best possible estimate for eachselected characteristic of the monitored object, to provide the accuracyof the estimate, and to significantly improve the temporal resolutionare significant advantages of the present powered weightedincrements-based method.

The present system for measuring characteristics of continuous mediumand/or localized targets can also include a quality control algorithm(s)either at all steps 102-106 in FIG. 1 or at selected steps if onechooses to apply this option. If quality control is applied, one canalso get a measure of the reliability for the best estimate at this step106 using appropriate procedures. It is important that the measurementaccuracy and the reliability are completely different values. Theaccuracy is related to a random scatter between multiple estimates. Thereliability is related to systematic errors in all multiple estimatesfor a given characteristic due to noise, outliers, sensormalfunctioning, and so on, for example, the confidence in the case of afuzzy logic based quality control algorithm.

(b) A joint statistical analysis of the selected characteristics of themonitored object at an analyzed averaging time interval T_(av) and rangeR with those from a previous interval(s) for the same range if onechooses to apply this option. Such an analysis can be accomplished byusing all estimates for each particular selected characteristic of theobject, statistics of estimates from the previous procedure listed in(a), or any combination thereof.

(c) A joint statistical analysis of the selected characteristics of themonitored object at an analyzed averaging time interval T_(av) and rangeR with those from other close enough ranges if one chooses to apply thisoption. Such an analysis can be accomplished by using all estimates foreach particular selected characteristic of the object, statistics ofestimates from the previous procedure(s) (a) and/or (b), or anycombination of the above.

(d) An identification of the monitored object in accordance withpredetermined requirements by using a set of measured indicators if suchan identification is required. Such identification can be accomplishedby using all estimates for each predetermined indicator, statistics ofestimates from the previous procedure(s) (a) and/or (b) and/or (c), orany combination of the above.

User Display and/or Data Transfer and/or Archiving

The best and/or multiple estimates for the selected characteristics ofthe monitored object produced by the present system and method formeasuring characteristics of continuous medium and/or localized targetscan be displayed at step 107 shown in FIG. 1 in any user specifiedformat. This may be range-time arrows or bars, range-time color-codedplots, time series of selected characteristics for selected ranges, andother well-known formats. The user-specified set of characteristics canalso be transferred to specified remote locations and/or archived forfuture use and/or reference. One can transfer or/and archive allestimates for each selected characteristic, the best estimates, or anycombination of the above. Archiving and/or transfer can be accomplishedin any user-specified data format and into any type of local or remotedata storage (the computer hard disc, CD-ROM, tape, etc.).

Necessary Conditions for Using the Present Invention

The present system and method for measuring characteristics ofcontinuous medium and/or localized targets is based on calculating andanalyzing powered weighted increments for pairs of signals. The presentinvention requires the powered weighted increments to be presented asmathematical models with adjustable parameters where the parameters arerelated to the selected characteristics of the monitored object. Themodels for the powered weighted increments can be constructed with thepresent method under one major requirement which imposes theoretical andpractical limitations on applying the present system and method tospecific monitoring equipment.

Requirement N1: Multiple Sensors

The powered weighted increments are defined by Eq. (9) for receivedsignals from a pair of physical and/or virtual sensors A_(m1) and A_(m2)with spatially separated centers {right arrow over (x)}_(Am1) and {rightarrow over (x)}_(Am2), hence at least two physical or virtual sensorsmust be utilized. However, the present system and method enable one tomeasure characteristics of the monitored object in only one directionδ{right arrow over (x)}_(m2,m1)={right arrow over (x)}_(Am2)−{rightarrow over (x)}_(Am1); see, for example equations (30), (35), (42), and(45). If the selected characteristics of the monitored object arerelated to multiple directions, the deployed physical and/or virtualsensors must provide at least one baseline in each of the directions.For example, three physical sensors with spatially separated phasecenters in FIG. 2B provide three non-parallel baselines in thehorizontal plane which allows for measuring all of the selectedcharacteristics of the atmosphere in any specified horizontal direction.To get characteristics of the atmosphere in the vertical direction, forexample the mean vertical velocity <W>, at least one sensor in FIG. 2Bshould be positioned at a different height with respect to othersensors. Each additional sensor provides additional information aboutthe monitored object, increases the accuracy and reliability of theestimates, improves the system's redundancy to account for the failureof some sensors, or any combination of the above features.

Preferred Conditions for Using the Present Invention

The preferred mathematical models for the powered weighted increments inthe present invention are analytical operational equations in the formof decompositions into polynomial functions over sufficiently smalltemporal and/or spatial separations where adjustable parameters are thecoefficients in the decompositions. The preferred models can beconstructed with the present method under two conditions listed below.

Condition P1: Small Temporal and/or Spatial Separations

The powered weighted increments (3), (9) are the most efficient dataprocessing tool when they contain at least one small parameter: asufficiently small temporal separation τ between the obtained signalsand/or a sufficiently small spatial separation |δ{right arrow over(x)}_(m2,m1)|=|{right arrow over (x)}_(Am2)−{right arrow over(x)}_(Am1)| between the sensors. In this case the mathematical modelsfor the increments can be derived in the preferred form ofdecompositions into polynomial functions, for example, into the Taylorseries (10) or (10a). The coefficients in the decompositions are theadjustable parameters which can be analytically related to the selectedcharacteristics of the monitored object with the present method in mostpractical cases.

In the given example of the atmospheric profiling radar based system, asufficiently small temporal separation τ can always be achieved byapplying τ=0, ±δt, and ±2δt at a very small δt. The powered weightedincrements (9) can then be decomposed into the Tailor series (10) at τ→0and the coefficients in the decomposition can be related to the selectedcharacteristics of the atmosphere by using analytical expressions suchas equations (30), (35), (39), (42), and (45). In general, specificconditions on the magnitude of δt depend on the type, configuration, andoperational mode of the monitoring equipment as well as on thecharacteristics of the object to be determined. In the given example ofan atmospheric profiler, δt should be small enough to ensure asufficiently small |ψ_(i)| such as |ψ_(i)|²<<|ψ_(i)|, for example,|ψ_(i)|<⅛. It can be formalized in a conservative way as follows:

$\begin{matrix}\begin{matrix}{{{\delta\; t} < {\frac{1}{64\pi\;\gamma}\frac{D}{{V_{h}}_{\max}}}},} & {{\delta\; t} < {\frac{1}{{64\pi}\;}\frac{\lambda}{{W}_{\max}}}}\end{matrix} & (51)\end{matrix}$Here |V_(h)|_(max) and |W|_(max) are the maximum expected values of therespective horizontal and vertical velocities in the atmosphere. Themaximum expected velocities in the atmospheric boundary layer are|V_(h)|_(max)≈50 m/s and |W|_(max)≈3 m/s, and typical parameters of theboundary layer profilers are D≧2 m, λ≧33 cm, and γ≈0.4. It then followsfrom Eq. (51) that the sampling time interval δt<0.5 ms ensuresefficient application of the present system and method at suchconditions. Atmospheric boundary layer profiling radars typicallyoperate at a pulse repetition frequency PRF=10 KHz or higher whichcorresponds to δt≦0.1 ms, therefore condition (51) can always be easilysatisfied by choosing the appropriate number of coherent integrations.Condition P2: Correlated Signals

The powered weighted increments are the most efficient data processingtool when applied to highly correlated (but not identical) signals; thisrequirement can be formalized as follows:

$\begin{matrix}{{\rho_{\min} \leq {\rho\left( {\delta\;{\overset{->}{x}}_{{m\; 2},{m\; 1}}} \right)}} = {\frac{\left\langle {\left\lbrack {{S\left( {{\overset{->}{x}}_{{Am}\; 1},t} \right)} - \left\langle {S\left( {{\overset{->}{x}}_{{Am}\; 1},t} \right)} \right\rangle} \right\rbrack\left\lbrack {{S\left( {{\overset{->}{x}}_{{Am}\; 2},t} \right)} - \left\langle {S\left( {{\overset{->}{x}}_{{Am}\; 2},t} \right)} \right\rangle} \right\rbrack} \right\rangle}{\left\langle \left\lbrack {{S\left( {{\overset{->}{x}}_{{Am}\; 1},t} \right)} - \left\langle {S\left( {{\overset{->}{x}}_{{Am}\; 1},t} \right)} \right\rangle} \right\rbrack^{2} \right\rangle} \leq \rho_{\max}}} & (52)\end{matrix}$Here ρ(δ{right arrow over (x)}_(m2,m1)) is the correlation coefficientbetween the actual signals from sensors A_(m1) and A_(m2) with the phasecenters {right arrow over (x)}_(Am1) and {right arrow over (x)}_(Am2),and ρ_(min)≈0.3, ρ_(max)≈0.99 define an efficient operational range forthe present system. Specific limitations on monitoring equipment thatensure the desirable correlation among the signals depend on the type,configuration, and operational mode of the equipment as well as thecharacteristics of the object to be determined. In the given example ofthe atmospheric profiling radar based system, the condition for signalsto be correlated imposes specific limitations on the spatial separationbetween the phase centers of the physical sensors δ{right arrow over(x)}_(m2,m1) shown in FIG. 2B. One can relate the powered weightedincrements and the correlation coefficient as follows:ρ(δ{right arrow over (x)} _(m2,m1))=1−d ₀(δ{right arrow over (x)}_(m2,m1),0,ω_(τ))/2   (53)One can get a specific limitation on the separation δ{right arrow over(x)}_(m2,m1) for the given example using equations (52), (53), and (34)and (53) to obtain the following inequality:

$\begin{matrix}{{\frac{- {\ln\left( \rho_{\max} \right)}}{2{\pi\gamma}}D} \leq {{\delta\;{\overset{->}{x}}_{{m\; 2},{m\; 1}}}} \leq {\frac{- {\ln\left( \rho_{\min} \right)}}{2{\pi\gamma}}D}} & (54)\end{matrix}$This condition can always be satisfied for any atmospheric profilingradar, for example.Alternative Implementations of the Present Invention

The monitoring equipment can deploy in-situ sensors (temperature,pressure, concentration, etc.), passive remote sensors (arrays ofmicrophones used for wake vortices detection, radio telescopes,radiometers, etc.), or active remote sensors (radars, sonars, soundgenerators combined with arrays of microphones such as for undergroundexploration, etc.) and still fall within the scope and intent of thepresent invention.

The deployed sensors can be of the same type (measuring the sameparameters of the monitored object as in spaced antenna radars) ordifferent types (temperature, pressure, and/or concentration sensors,etc.), can have any shape (hexagon, square, triangle, circle, etc.antennas) and any construction (mechanical, electronic, etc.), and canoperate at any wavelength and still fall within the scope and intent ofthe present invention.

The monitoring equipment can be configured to deploy any number ofsensors (two or more) spatially separated in at least one spatialdimension; fully separated sensors, adjacent sensors (as in FIG. 2B), oroverlapping sensors (as in FIG. 4). The deployed sensors can be fullypassive (as in-situ temperature probes or radio telescope antennas),fully or partly used for transmitting (as in FIG. 2), or be fullyseparated from a transmitter; and still fall within the scope and intentof the present invention.

The monitoring equipment can be mounted on a fixed platform (ground,tower, building, etc.), or on a moving platform (truck, ship, airplane,satellite, etc.) and still fall within the scope and intent of thepresent invention.

The deployed sensors can operate in a pulse or continuous wave mode, ascanning or fixed direction mode, and the processing can be executed inreal-time or in off-field mode and still fall within the scope andintent of the present invention.

An actual signal can be the complex output from a quadrature-phasesynchronous detector, the signal power, the signal amplitude, or thesignal phase, and the combined signals can be any function of the actualsignals and still fall within the scope and intent of the presentinvention.

The averaging for calculating the powered weighted increments can beexecuted over an ensemble of observations, over any specified timeinterval (including no temporal averaging for locally non-stationaryprocesses), over any specified spatial domain, or any combination of theabove, and still fall within the scope and intent of the presentinvention.

The mathematical models with adjustable parameters for the poweredweighted increments can be analytical expressions, tabulated results ofnumerical simulations or experiments, and the like. The relationshipsbetween the adjustable parameters and the selected characteristics ofthe monitored object can depend on the type, configuration, andoperational mode of the monitoring equipment, and/or the underlyingtheory and techniques that are chosen for constructing the models of theobject and the characteristics of the object to be determined. Anycombination of these aspects related to modeling should be considered tofall within the scope and intent of the present invention.

Exemplary Embodiment of the System

FIG. 5 is a block diagram showing a preferred embodiment of the system200 for obtaining data indicative of selected characteristics of amonitored object within a predetermined volume of space. The system 200includes a sensor configuration 220 for acquiring a plurality ofsignals. The sensor configuration 220 can be configured and operated inany of the ways described above with reference to FIGS. 2A, 2B, 3, 4A,4B, and 4C. The system 200 also includes a processing circuit 240configured for obtaining data indicative of the selected characteristicsof the monitored object by calculating a plurality of powered weightedincrements using the plurality of signals from the sensor configuration220 and using a plurality of models for relating the plurality ofpowered weighted increments to the characteristic or characteristics ofthe medium or the target. The necessary functioning of the processingcircuit 240 has been described above with reference to steps 102-106shown in FIG. 1. It should be clear to one of ordinary skill in the artthat the processing circuit 240 can be constructed using anycommercially available microprocessor, any suitable combination ofdigital or analog electronic components, or using a computer. Using anoutput circuit 260, the data indicative of the selected characteristicsof the monitored object(s) is then output to be stored in a data storagedevice, to be transferred to a remote location, and/or to be output in asuitable format to a display.

1. A system for obtaining data indicative of at least one characteristicof a continuous medium or at least one localized target located within apredetermined volume of space, which comprises: a sensor configurationincluding a plurality of sensors for acquiring a plurality of signalsfrom the continuous medium or the target, said plurality of sensorshaving centers spatially separated from each other in at least onespatial dimension; and a processing circuit for obtaining said dataindicative of said characteristic of the continuous medium or the targetby calculating a plurality of powered weighted increments using saidplurality of signals acquired by said sensor configuration and byrelating said plurality of powered weighted increments to saidcharacteristic of the continuous medium or the target using a pluralityof models.
 2. The system according to claim 1, wherein said plurality ofsensors is configured for concurrently acquiring said plurality ofsignals from the continuous medium or the target.
 3. The systemaccording to claim 2, wherein each one of said plurality of sensors ispositioned at a predetermined location inside the predetermined volumeof space.
 4. The system according to claim 2, wherein: said plurality ofsensors is positioned outside the predetermined volume of space; andsaid plurality of signals acquired by said plurality of sensors isgenerated by the continuous medium or the target.
 5. The systemaccording to claim 2, wherein: said plurality of sensors is positionedoutside the predetermined volume of space; and said plurality of signalsacquired by said plurality of sensors is caused by predeterminedradiation that is generated and propagated through the predeterminedvolume of space to induce backscatter from the continuous medium or thetarget.
 6. The system according to claim 1, wherein said processingcircuit is configured for increasing an amount of informationextractable from said plurality of signals by modifying said pluralityof signals, and subsequently obtaining said data indicative of saidcharacteristic of the continuous medium or the target.
 7. The systemaccording to claim 6, wherein said processing circuit is configured formodifying said plurality of signals by performing at least onemodification step selected from a group consisting of: converting saidplurality of signals from complex signals to real signals, removingnoise from said plurality of signals, removing mean values from saidplurality of signals, normalizing each one of said plurality of signalswith a standard deviation of the respective one of said plurality ofsignals, and generating virtual sensors using combinations of saidplurality of signals.
 8. The system according to claim 1, wherein saidprocessing circuit is configured for calculating said plurality ofpowered weighted increments for one or more specified orders, specifiedpairs of signals from said plurality of said sensors, and specifiedcombinations of weights.
 9. The system according to claim 1, whereinsaid processing circuit is configured for relating said plurality ofpowered weighted increments to said characteristic of the medium ortarget by: fitting said plurality of powered weighted increments to aplurality of predetermined models, estimating a plurality of adjustableparameters in said plurality of predetermined models, and relating saidplurality of adjustable parameters to said characteristic of thecontinuous medium or the target.
 10. The system according to claim 9,wherein at least one of said plurality of predetermined models is formedas a decomposition into a Taylor series.
 11. The system according toclaim 9, wherein each one of said plurality of predetermined models isconstructed from at least one model selected from a group consisting ofan analytically derived operational equation formed as a decompositioninto a polynomial function over a selected parameter, a tabulatedfunction obtained using a numerical simulation, and a tabulated functionobtained using a physical experiment.
 12. The system according to claim1, wherein said processing circuit is configured for increasing anaccuracy of said data indicative of said characteristic of the medium orthe target by analyzing multiple estimates of said data indicative ofsaid characteristic of the continuous medium or the target.
 13. Thesystem according to claim 1, further comprising an output circuit foroutputting said data indicative of said characteristic of the continuousmedium or the target.
 14. A method for obtaining data indicative of atleast one characteristic of a continuous medium or at least onelocalized target located within a predetermined volume of space, whichcomprises: using a sensor configuration having a plurality of sensorswith centers spatially separated from each other in at least one spatialdimension to acquire a plurality of signals from the continuous mediumor the target; and obtaining the data indicative of the characteristicof the continuous medium or the target by calculating a plurality ofpowered weighted increments using the plurality of signals acquired bythe plurality of sensors and by relating the plurality of poweredweighted increments to the characteristic of the medium or the targetusing a plurality of models.
 15. The method according to claim 14, whichcomprises concurrently acquiring the plurality of signals from thecontinuous medium or the target with the plurality of sensors.
 16. Themethod according to claim 14, which comprises positioning each one ofthe plurality of sensors at a predetermined location inside thepredetermined volume of space.
 17. The method according to claim 14,which comprises positioning each one of the plurality of sensors at apredetermined location outside the predetermined volume of space; theplurality of signals acquired by the plurality of sensors beinggenerated by the continuous medium or the target.
 18. The methodaccording to claim 14, which comprises: positioning each one of theplurality of sensors at a predetermined location outside thepredetermined volume of space; and generating and propagatingpredetermined radiation through the predetermined volume of space toinduce backscatter from the continuous medium or the target in a mannerenabling the plurality of signals to be acquired by the plurality ofsensors.
 19. The method according to claim 14, which comprises:increasing an amount of information extractable from the plurality ofsignals by modifying the plurality of signals; and subsequentlyperforming the step of obtaining the data indicative of thecharacteristic of the continuous medium or the target.
 20. The methodaccording to claim 14, which comprises: increasing an amount ofinformation extractable from the plurality of signals by performing atleast one modification step selected from a group consisting of:converting the plurality of signals from complex signals to realsignals, removing noise from the plurality of signals, removing meanvalues from the plurality of signals, normalizing each one of theplurality of signals with a standard deviation of the respective one ofthe plurality of signals, and generating virtual sensors usingcombinations of the plurality of signals; and subsequently performingthe step of obtaining the data indicative of the characteristic of thecontinuous medium or the target.
 21. The method according to claim 14,which comprises calculating the plurality of powered weighted incrementsfor one or more specified orders, specified pairs of signals from theplurality of the sensors, and specified combinations of weights.
 22. Themethod according to claim 14, which comprises relating the plurality ofpowered weighted increments to the characteristic of the medium ortarget by: fitting the plurality of powered weighted increments to aplurality of predetermined models, estimating a plurality of adjustableparameters in the plurality of predetermined models, and relating theplurality of adjustable parameters to the characteristic of thecontinuous medium or the target.
 23. The method according to claim 22,which comprises forming at least one of the plurality of predeterminedmodels as a decomposition into a Taylor series.
 24. The method accordingto claim 22, which comprises constructing each one of the plurality ofpredetermined models from at least one model selected from a groupconsisting of: an analytically derived operational equation formed as adecomposition into a polynomial function over a selected parameter, atabulated function obtained using a numerical simulation, and atabulated function obtained using a physical experiment.
 25. The methodaccording to claim 14, which comprises increasing an accuracy of thedata indicative of the characteristic of the continuous medium or thetarget by analyzing multiple estimates of the data indicative of thecharacteristic of the continuous medium or the target.
 26. The methodaccording to claim 14, which comprises outputting the data indicative ofthe characteristic of the continuous medium or the target.